Chapter 5 Lines and Angles (Additional Questions)

Given:

Two angles are complementary.

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To Find:

The sum of the two complementary angles.

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Solution:

Two angles are said to be complementary if the sum of their measures is $90^\circ$.

Let the two complementary angles be $\angle A$ and $\angle B$.

According to the definition of complementary angles:

Therefore, the sum of two complementary angles is $90^\circ$.

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The correct option is (A) $90^\circ$.

Given:

An angle measuring $75^\circ$.

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To Find:

The supplement of the given angle.

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Solution:

Two angles are said to be supplementary if the sum of their measures is $180^\circ$.

Let the given angle be $\angle A = 75^\circ$.

Let the supplement of this angle be $\angle B$.

According to the definition of supplementary angles, the sum of the angle and its supplement must be $180^\circ$.

Substitute the given value of $\angle A$:

To find $\angle B$, subtract $75^\circ$ from both sides of the equation:

Performing the subtraction:

$180 - 75 = 105$

So, the supplement of $75^\circ$ is $105^\circ$.

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The correct option is (B) $105^\circ$.

Given:

Two adjacent angles form a linear pair.

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To Find:

The property of their non-common arms.

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Solution:

Adjacent angles are two angles that have a common vertex and a common arm, but no common interior points.

A linear pair is a pair of adjacent angles whose non-common arms are opposite rays.

The sum of the angles in a linear pair is always $180^\circ$ (they are supplementary).

For two adjacent angles to form a linear pair, their non-common arms must extend in opposite directions from the common vertex, forming a straight line.

This means the non-common arms are opposite rays.

Here, B is the common vertex and BC is the common arm.

For $\angle ABC$ and $\angle CBD$ to form a linear pair, the non-common arms BA and BD must be opposite rays.

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The correct option is (C) Opposite rays.

Given:

Two lines intersect.

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To Find:

The property of the vertically opposite angles formed.

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Solution:

When two straight lines intersect at a point, they form four angles.

The angles that are opposite to each other at the intersection point are called vertically opposite angles.

Let two lines AB and CD intersect at the point O, as shown below (imaginary diagram).

The pairs of vertically opposite angles formed are:

1. $\angle AOC$ and $\angle BOD$

2. $\angle COB$ and $\angle DOA$

A fundamental geometric theorem states that vertically opposite angles are always equal.

We can prove this using the property of linear pairs.

Angles forming a linear pair are supplementary, meaning their sum is $180^\circ$.

Consider line AB and ray OD. $\angle AOC$ and $\angle COB$ form a linear pair.

Consider line CD and ray OB. $\angle COB$ and $\angle BOD$ form a linear pair.

From equation (i) and equation (ii), we have:

Subtract $\angle COB$ from both sides of the equation:

This proves that one pair of vertically opposite angles is equal.

Similarly, we can show that $\angle COB = \angle DOA$.

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Thus, if two lines intersect, the vertically opposite angles formed are always equal.

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The correct option is (C) Equal.

Given:

An angle is $20^\circ$ less than its complement.

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To Find:

The measure of the angle.

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Solution:

Let the measure of the unknown angle be $x$ degrees.

Two angles are complementary if their sum is $90^\circ$.

The complement of the angle $x$ is $90^\circ - x$.

According to the problem statement, the angle ($x$) is $20^\circ$ less than its complement ($90^\circ - x$).

This can be written as an equation:

Now, we solve the equation for $x$:

Add $x$ to both sides of the equation:

Divide both sides by 2:

So, the measure of the angle is $35^\circ$.

Let's verify the answer:

The angle is $35^\circ$.

Its complement is $90^\circ - 35^\circ = 55^\circ$.

The difference between the complement and the angle is $55^\circ - 35^\circ = 20^\circ$.

This matches the condition given in the problem (the angle is $20^\circ$ less than its complement).

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The correct option is (A) $35^\circ$.

Given:

Four pairs of angles.

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To Find:

Which pair is NOT supplementary.

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Solution:

Two angles are said to be supplementary if the sum of their measures is $180^\circ$.

We need to find the sum of the angles in each given pair to determine which pair does not add up to $180^\circ$.

Let's check each option:

(A) $90^\circ$ and $90^\circ$:

This is a pair of supplementary angles.

(B) $60^\circ$ and $120^\circ$:

This is a pair of supplementary angles.

(C) $45^\circ$ and $135^\circ$:

This is a pair of supplementary angles.

(D) $30^\circ$ and $60^\circ$:

This sum is $90^\circ$, which is the definition of complementary angles, not supplementary angles.

Therefore, the pair $30^\circ$ and $60^\circ$ is NOT a pair of supplementary angles.

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The correct option is (D) $30^\circ$ and $60^\circ$.

Given:

A transversal intersects two lines.

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To Find:

The total number of angles formed by the intersection.

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Solution:

A transversal is a line that intersects two or more other lines at distinct points.

When a transversal line intersects a single line, it creates two angles on one side of the transversal and two angles on the other side, around the intersection point. This forms a total of 4 angles around that point.

In this problem, the transversal intersects two lines.

Let the transversal be line T, intersecting lines L1 and L2.

The transversal T intersects line L1 at one point.

At this intersection point, 4 angles are formed.

The transversal T intersects line L2 at another distinct point.

At this second intersection point, another 4 angles are formed.

The total number of angles formed is the sum of the angles formed at each intersection point.

These 8 angles include pairs of vertically opposite angles, linear pairs, corresponding angles, alternate interior angles, alternate exterior angles, consecutive interior angles, and consecutive exterior angles, depending on the properties of lines L1 and L2 (e.g., if they are parallel).

However, regardless of whether the two lines are parallel or not, a transversal intersecting two distinct lines will always create 8 angles in total.

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The correct option is (C) 8.

Given:

Two lines intersected by a transversal.

A description of the position of a pair of angles: on the same side of the transversal and both above the two lines.

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To Find:

The name of the angles that fit the given description.

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Solution:

When a transversal line intersects two other lines, eight angles are formed at the two intersection points.

These angles are given specific names based on their positions relative to the transversal and the two lines.

Let's consider the definitions of the angle pairs listed in the options:

Alternate interior angles: These are pairs of angles located on opposite sides of the transversal and between the two lines.

Corresponding angles: These are pairs of angles located on the same side of the transversal and in corresponding positions relative to the two lines. This means one angle is above the first line and the other is above the second line (on the same side of the transversal), OR one is below the first line and the other is below the second line (on the same side of the transversal).

Consecutive interior angles (or Same-side interior angles): These are pairs of angles located on the same side of the transversal and between the two lines.

Vertically opposite angles: These are pairs of non-adjacent angles formed by the intersection of two lines. They share a vertex but not an arm and are directly opposite each other. They are always equal.

The question describes angles that are "on the same side of the transversal" and "both above the two lines".

This description precisely matches the definition of Corresponding angles, where the angles are on the same side of the transversal and are in the "above" position relative to their respective lines.

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The correct option is (B) Corresponding angles.

Given:

Two parallel lines are intersected by a transversal.

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To Find:

Which pairs of angles formed are equal.

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Solution:

When a transversal intersects two lines, eight angles are formed. If the two lines are parallel, specific relationships exist between these angles.

Let the two parallel lines be $m$ and $n$, and let the transversal be $t$. The intersection forms 8 angles.

The properties of angles formed when a transversal intersects parallel lines are:

1. Corresponding Angles: Pairs of corresponding angles are equal.

For example, the angle in the upper left position at the first intersection is equal to the angle in the upper left position at the second intersection.

2. Alternate Interior Angles: Pairs of alternate interior angles are equal.

These are angles on opposite sides of the transversal and between the parallel lines.

3. Alternate Exterior Angles: Pairs of alternate exterior angles are equal.

These are angles on opposite sides of the transversal and outside the parallel lines.

4. Consecutive Interior Angles: Pairs of consecutive interior angles are supplementary (their sum is $180^\circ$).

These are angles on the same side of the transversal and between the parallel lines.

The question asks which pairs of angles are equal when the lines are parallel.

From the properties listed above, Corresponding angles, Alternate interior angles, and Alternate exterior angles are all equal when the lines are parallel.

Therefore, all the pairs mentioned in options (A), (B), and (C) are equal.

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The correct option is (D) All of the above.

Given:

Two adjacent angles whose sum is $180^\circ$.

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To Find:

The type of pair formed by these angles.

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Solution:

Let the two adjacent angles be $\angle A$ and $\angle B$.

Adjacent angles share a common vertex and a common arm.

We are given that their sum is $180^\circ$.

Let's examine the definitions provided in the options:

(A) Complementary pair: The sum of the angles is $90^\circ$. This does not match the given condition.

(B) Supplementary pair: The sum of the angles is $180^\circ$. This matches the given condition.

(C) Linear pair: A linear pair is a pair of adjacent angles whose non-common arms are opposite rays. When the non-common arms are opposite rays, they form a straight line, and the sum of the angles is always $180^\circ$. Conversely, if two adjacent angles sum to $180^\circ$, their non-common arms must lie on the same straight line, forming opposite rays. Thus, adjacent angles with a sum of $180^\circ$ also form a linear pair.

Since the given angles are adjacent and their sum is $180^\circ$, they satisfy the definition of both a supplementary pair and a linear pair.

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The correct option is (D) Both (B) and (C).

Given:

The measure of angle A is $50^\circ$.

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To Find:

The measure of the angle vertically opposite to angle A.

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Solution:

When two lines intersect, they form pairs of vertically opposite angles.

A fundamental geometric property is that vertically opposite angles are always equal in measure.

Let angle A be formed by the intersection of two lines. The angle vertically opposite to angle A is the angle across the intersection point from A, sharing only the vertex.

Let the angle vertically opposite to angle A be angle B.

According to the property of vertically opposite angles:

We are given that the measure of angle A is $50^\circ$.

Therefore, the measure of the angle vertically opposite to angle A is $50^\circ$.

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The correct option is (C) $50^\circ$.

Given:

A question asking for the condition that defines parallel lines.

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To Find:

The correct definition of parallel lines from the given options.

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Solution:

In Euclidean geometry, two lines are defined as parallel lines if they lie in the same plane and do not intersect, no matter how far they are extended in either direction.

Let's analyze the given options:

(A) Intersect at one point: This describes intersecting lines, not parallel lines.

(B) Meet at infinity: This is a concept often used in projective geometry. In the context of standard Euclidean geometry, parallel lines do not meet at all.

(C) Never intersect and the distance between them remains constant: This is the precise definition of parallel lines in Euclidean geometry. The property of never intersecting implies that the perpendicular distance between the two lines remains the same at all points along their length.

(D) Are perpendicular to each other: This describes lines that intersect at a right angle ($90^\circ$). Perpendicular lines are a specific type of intersecting lines, not parallel lines.

Based on the definition, the condition that correctly identifies parallel lines is that they never intersect and maintain a constant distance from each other.

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The correct option is (C) Never intersect and the distance between them remains constant.

Given:

A transversal intersects two parallel lines.

One interior angle on the same side of the transversal measures $110^\circ$.

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To Find:

The measure of the other interior angle on the same side of the transversal.

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Solution:

When a transversal intersects two parallel lines, the pairs of consecutive interior angles (also known as same-side interior angles) are supplementary.

Supplementary angles are two angles whose measures sum up to $180^\circ$.

Let the measure of the given interior angle on one side of the transversal be $\angle 1 = 110^\circ$.

Let the measure of the other interior angle on the same side of the transversal be $\angle 2$.

According to the property of consecutive interior angles formed by a transversal intersecting parallel lines:

Substitute the given value of $\angle 1$ into the equation:

To find $\angle 2$, subtract $110^\circ$ from both sides of the equation:

Performing the subtraction:

$180 - 110 = 70$

So, the measure of the other interior angle on the same side is $70^\circ$.

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The correct option is (B) $70^\circ$.

Given:

Options describing different pairs of angles formed by lines and a transversal.

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To Find:

The angle pair that is always supplementary.

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Solution:

Supplementary angles are two angles whose measures add up to $180^\circ$.

Let's analyze the properties of each angle pair listed in the options when a transversal intersects two lines (assuming the context relates to parallel lines where specified):

(A) Corresponding angles (when lines are parallel): When two parallel lines are intersected by a transversal, corresponding angles are equal, not supplementary (unless they are both $90^\circ$, which is a specific case, not 'always').

(B) Alternate interior angles (when lines are parallel): When two parallel lines are intersected by a transversal, alternate interior angles are equal, not supplementary (unless they are both $90^\circ$, a specific case).

(C) Vertically opposite angles: Vertically opposite angles are formed by intersecting lines. They are always equal, not supplementary (unless they are both $90^\circ$). This property holds regardless of whether the lines are parallel or not, as it depends only on the intersection of the two lines forming the angles.

(D) Consecutive interior angles (when lines are parallel): When two parallel lines are intersected by a transversal, consecutive interior angles (angles on the same side of the transversal and between the parallel lines) are always supplementary. Their sum is always $180^\circ$.

Based on the definitions and properties, the pair of angles that is always supplementary (when the lines are parallel, as specified in the option) is the consecutive interior angles.

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The correct option is (D) Consecutive interior angles (when lines are parallel).

Given:

A list of angle types formed by two lines and a transversal, and a list of properties/descriptions.

Assumption: The two lines are parallel.

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To Find:

The correct matching of angle types with their properties/descriptions.

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Solution:

Let's analyze each angle type and match it with the appropriate property or description from the second list:

(i) Corresponding angles: These angles are on the same side of the transversal, one is interior (between the two lines), and the other is exterior (outside the two lines). They are in corresponding positions (e.g., both above the lines or both below the lines). This description matches option (d).

When the two lines are parallel, corresponding angles are also equal, which matches option (b). However, option (d) provides a positional description, while (b) is a property that holds specifically when the lines are parallel. Looking at the options, (i) is paired with (d) in option (A), which seems like a positional match.

(ii) Alternate interior angles: These angles are on opposite sides of the transversal and are both interior (between the two lines). When the two lines are parallel, alternate interior angles are always equal. This property matches option (b).

(iii) Consecutive interior angles: These angles are on the same side of the transversal and are both interior (between the two lines). When the two lines are parallel, consecutive interior angles are supplementary, meaning their sum is $180^\circ$. This property matches option (a).

(iv) Vertically opposite angles: These angles are formed by the intersection of two lines. They are directly opposite each other and share a common vertex. Vertically opposite angles are always equal. The description "Formed by intersecting lines" directly relates to how they are created, which matches option (c).

Based on this analysis, the correct matches are:

• (i) Corresponding angles $\rightarrow$ (d) On the same side of transversal, one interior, one exterior
• (ii) Alternate interior angles $\rightarrow$ (b) Equal (when lines are parallel)
• (iii) Consecutive interior angles $\rightarrow$ (a) Sum is $180^\circ$ (when lines are parallel)
• (iv) Vertically opposite angles $\rightarrow$ (c) Formed by intersecting lines

Let's compare these matches to the provided options:

• (A) (i)-(d), (ii)-(b), (iii)-(a), (iv)-(c) - This matches our derived pairings.
• (B) (i)-(b), (ii)-(d), (iii)-(a), (iv)-(c) - (i)-(b) is a property when parallel, but (d) is a positional description often used to identify them. (ii)-(d) is incorrect as alternate interior angles are not described this way.
• (C) (i)-(d), (ii)-(a), (iii)-(b), (iv)-(c) - (ii)-(a) is incorrect (alternate interior are equal, not supplementary). (iii)-(b) is incorrect (consecutive interior are supplementary, not equal).
• (D) (i)-(d), (ii)-(b), (iii)-(c), (iv)-(a) - (iii)-(c) is incorrect (consecutive interior are not defined by how they are formed relative to intersection lines in this context). (iv)-(a) is incorrect (vertically opposite angles are equal, not supplementary).

Therefore, option (A) provides the correct matches.

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The correct option is (A) (i)-(d), (ii)-(b), (iii)-(a), (iv)-(c).

Given:

Assertion (A): If two angles form a linear pair, they are supplementary.

Reason (R): Supplementary angles sum up to $180^\circ$.

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To Find:

Whether Assertion (A) and Reason (R) are true or false, and if R is the correct explanation for A.

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Solution:

Let's evaluate the Assertion (A) and the Reason (R) separately.

Assertion (A): If two angles form a linear pair, they are supplementary.

A linear pair consists of two adjacent angles whose non-common arms are opposite rays. When two rays are opposite, they form a straight line, which has a measure of $180^\circ$. The two adjacent angles of a linear pair divide this straight angle. Therefore, the sum of the measures of the angles in a linear pair is always $180^\circ$. Angles whose sum is $180^\circ$ are defined as supplementary angles.

Thus, Assertion (A) is True.

Reason (R): Supplementary angles sum up to $180^\circ$.

This statement is the definition of supplementary angles.

Thus, Reason (R) is True.

Now, let's determine if Reason (R) is the correct explanation for Assertion (A).

Assertion (A) states that linear pairs are supplementary. This is true because the sum of angles in a linear pair is $180^\circ$. Reason (R) provides the definition of supplementary angles, which is exactly the condition that makes a linear pair supplementary. The fact that supplementary angles sum to $180^\circ$ is the fundamental property that explains why a linear pair, having a sum of $180^\circ$, fits the definition of supplementary angles.

Therefore, Reason (R) correctly explains Assertion (A).

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Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of Assertion (A).

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The correct option is (A) Both A and R are true, and R is the correct explanation of A.

Given:

Assertion (A): If a transversal intersects two lines such that corresponding angles are equal, then the two lines are parallel.

Reason (R): The corresponding angles property is a test for parallel lines.

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To Find:

Whether Assertion (A) and Reason (R) are true or false, and if R is the correct explanation for A.

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Solution:

Let's evaluate the truthfulness of Assertion (A) and Reason (R).

Assertion (A): If a transversal intersects two lines such that corresponding angles are equal, then the two lines are parallel.

This statement is the converse of the Corresponding Angles Axiom/Theorem. The Corresponding Angles Axiom states that if two parallel lines are intersected by a transversal, then corresponding angles are equal. The converse of this theorem is a fundamental result in Euclidean geometry, which states that if the corresponding angles formed by a transversal are equal, then the lines intersected by the transversal must be parallel.

Thus, Assertion (A) is True.

Reason (R): The corresponding angles property is a test for parallel lines.

In geometry, a "test for parallel lines" is a condition or property that can be used to determine if two lines are parallel. The converse theorems related to angles formed by a transversal intersecting two lines provide these tests. Specifically, if any of the following conditions hold, then the two lines are parallel:

• Corresponding angles are equal.
• Alternate interior angles are equal.
• Consecutive interior angles are supplementary (sum to $180^\circ$).

The statement in Reason (R) correctly identifies the property mentioned in Assertion (A) (the corresponding angles property) as a criterion or test for determining parallel lines.

Thus, Reason (R) is True.

Now, let's consider if Reason (R) is the correct explanation for Assertion (A).

Assertion (A) states that the equality of corresponding angles implies parallelism. Reason (R) states that the corresponding angles property serves as a test for parallel lines. The fact that the corresponding angles property is a recognized test for parallel lines directly justifies why the statement in Assertion (A) is true. The reason explains the significance and validity of the condition mentioned in the assertion.

Therefore, Reason (R) is the correct explanation for Assertion (A).

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Both Assertion (A) and Reason (R) are true, and Reason (R) correctly explains Assertion (A).

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The correct option is (A) Both A and R are true, and R is the correct explanation of A.

Given:

A figure showing two lines $l$ and $m$ intersected by a transversal $t$, with angles numbered 1 to 8.

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To Find:

Identify the pair of alternate exterior angles from the given options.

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Solution:

Let's define Alternate Exterior Angles:

Alternate exterior angles are a pair of angles formed when a transversal intersects two lines. They are located on opposite sides of the transversal and are outside the two lines.

In the given figure, lines $l$ and $m$ are intersected by transversal $t$.

The angles outside the lines $l$ and $m$ are $\angle 1, \angle 2, \angle 7, \angle 8$.

The transversal $t$ separates these exterior angles into two pairs based on which side of the transversal they are on.

The angles on one side of the transversal and outside the lines are $\angle 1$ and $\angle 7$.

The angles on the other side of the transversal and outside the lines are $\angle 2$ and $\angle 8$.

Alternate exterior angles must be on opposite sides of the transversal and both exterior.

The pairs of alternate exterior angles are:

1. $\angle 1$ (on the left of $t$, outside $l$) and $\angle 8$ (on the right of $t$, outside $m$).

2. $\angle 2$ (on the right of $t$, outside $l$) and $\angle 7$ (on the left of $t$, outside $m$).

Now let's check the given options against these pairs:

(A) $\angle 1$ and $\angle 7$: These are both on the left side of the transversal. So they are not alternate exterior angles. (They are consecutive exterior angles).

(B) $\angle 1$ and $\angle 8$: $\angle 1$ is on the left of $t$ and outside. $\angle 8$ is on the right of $t$ and outside. This pair fits the definition of alternate exterior angles.

(C) $\angle 3$ and $\angle 6$: These angles are between the lines (interior) and on opposite sides of the transversal. This pair represents alternate interior angles.

(D) $\angle 4$ and $\angle 5$: These angles are between the lines (interior) and on the same side of the transversal. This pair represents consecutive interior angles.

Therefore, the pair $\angle 1$ and $\angle 8$ are alternate exterior angles.

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The correct option is (B) $\angle 1$ and $\angle 8$.

The question asks which statement is true if lines $l$ and $m$ are parallel, based on a case study from Question 18. The case study is not provided, so we will analyze the options based on the standard properties of angles formed when a transversal intersects two parallel lines. We will assume a standard numbering convention for the angles formed by the transversal, where angles $\angle 1, \angle 2, \angle 3, \angle 4$ are formed at one intersection point (say, on line $l$) and angles $\angle 5, \angle 6, \angle 7, \angle 8$ are formed at the other intersection point (on line $m$). Let's typically number them such that at the first intersection, $\angle 1$ is top-left, $\angle 2$ is top-right, $\angle 3$ is bottom-right, and $\angle 4$ is bottom-left. At the second intersection, $\angle 5$ is top-left, $\angle 6$ is top-right, $\angle 7$ is bottom-right, and $\angle 8$ is bottom-left.

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When two parallel lines are intersected by a transversal, the following properties hold true:

• Corresponding Angles are equal (e.g., $\angle 1 = \angle 5$).
• Alternate Interior Angles are equal (e.g., $\angle 3 = \angle 6$).
• Alternate Exterior Angles are equal (e.g., $\angle 1 = \angle 8$).
• Consecutive Interior Angles (or Co-interior Angles) are supplementary, meaning their sum is $180^\circ$ (e.g., $\angle 3 + \angle 5 = 180^\circ$).

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Let's examine each given statement based on the standard numbering convention described above:

(A) $\angle 3 + \angle 6 = 180^\circ$

In the standard numbering, $\angle 3$ (bottom-right at the first intersection) and $\angle 6$ (top-right at the second intersection) are alternate interior angles. If lines $l$ and $m$ are parallel, alternate interior angles are equal, i.e., $\angle 3 = \angle 6$. Their sum $\angle 3 + \angle 6 = 180^\circ$ would only be true if $\angle 3 = \angle 6 = 90^\circ$. This is not true for all parallel lines. Therefore, statement (A) is generally false based on the standard numbering.

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(B) $\angle 1 = \angle 6$

In the standard numbering, $\angle 1$ (top-left at the first intersection) and $\angle 6$ (top-right at the second intersection) are not a standard pair (like corresponding or alternate interior). However, we can deduce a relationship. $\angle 1$ is vertically opposite to $\angle 3$ ($\angle 1 = \angle 3$). If lines $l$ and $m$ are parallel, $\angle 3$ is alternate interior to $\angle 6$ ($\angle 3 = \angle 6$). By the transitive property of equality, if $\angle 1 = \angle 3$ and $\angle 3 = \angle 6$, then $\angle 1 = \angle 6$. Thus, statement (B) is true when lines $l$ and $m$ are parallel under the standard numbering.

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(C) $\angle 4 = \angle 8$

In the standard numbering, $\angle 4$ (bottom-left at the first intersection) and $\angle 8$ (bottom-left at the second intersection) are corresponding angles. If lines $l$ and $m$ are parallel, corresponding angles are equal ($\angle 4 = \angle 8$). Thus, statement (C) is true when lines $l$ and $m$ are parallel under the standard numbering.

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(D) $\angle 2 + \angle 5 = 180^\circ$

In the standard numbering, $\angle 2$ (top-right at the first intersection) and $\angle 5$ (top-left at the second intersection) are not a standard pair whose sum is $180^\circ$. However, we can deduce a relationship. $\angle 2$ is corresponding to $\angle 6$ ($\angle 2 = \angle 6$). $\angle 5$ and $\angle 6$ form a linear pair, so their sum is $180^\circ$ ($\angle 5 + \angle 6 = 180^\circ$). Substituting $\angle 6 = \angle 2$ into the second equation, we get $\angle 5 + \angle 2 = 180^\circ$. Thus, statement (D) is true when lines $l$ and $m$ are parallel under the standard numbering.

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Based on the standard numbering convention, statements (B), (C), and (D) are all true when lines $l$ and $m$ are parallel. In a typical multiple-choice question, only one option should be correct. This suggests that the angle numbering specified in the case study of Question 18 is crucial and likely defines the angles such that only one of these statements is true. Without the specific diagram from Question 18, it is impossible to definitively determine the single intended correct answer.

However, if we assume that one of the options directly represents a fundamental property (Corresponding angles are equal, Alternate Interior angles are equal, or Consecutive Interior angles are supplementary), and considering option (B) involves a derived relationship under standard numbering whereas (C) directly involves corresponding angles (a fundamental type), and (A) and (D) involve sums ($180^\circ$), which could represent consecutive interior angles depending on numbering, the question is ambiguous.

Assuming the question intends to test a fundamental property and given the options, let's reconsider the standard numbering results: (A) False, (B) True (derived), (C) True (Corresponding), (D) True (derived). Since (C) directly states the equality of corresponding angles, which is a primary property, it is a strong candidate for the intended answer, provided the diagram in Q18 indeed shows $\angle 4$ and $\angle 8$ as corresponding angles.

Let's assume the question, despite the ambiguity under standard numbering, has a single correct answer among the choices provided. Without the diagram, selecting one answer definitively is challenging. However, if we interpret the options as representing relationships between specific pairs of angles *as labeled in the Q18 diagram*, we can infer what type of angles those pairs might be. For example, if option (B) $\angle 1 = \angle 6$ is the sole correct answer, it implies that $\angle 1$ and $\angle 6$ are likely alternate interior or corresponding angles in the Q18 diagram. Similarly for other options.

Let's assume, for the purpose of providing an answer, that the numbering in the diagram from Question 18 is such that option (B) represents the equality of alternate interior angles or corresponding angles.

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Let's assume the numbering in the missing diagram defines $\angle 1$ and $\angle 6$ such that they are alternate interior angles. If lines $l$ and $m$ are parallel, then alternate interior angles are equal.

If $\angle 1$ and $\angle 6$ are alternate interior angles as per the Q18 diagram, then statement (B) is true when $l \parallel m$. The other options would then be false based on the specific numbering in that diagram.

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Given the potential ambiguity without the diagram, and acknowledging that multiple options are true under standard numbering, we select option (B) assuming the diagram defines $\angle 1$ and $\angle 6$ as alternate interior angles.

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The final answer is $\boxed{B}$.

The question asks for the measure of an angle that is equal to its supplement.

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Let the measure of the angle be $\theta$ degrees.

Two angles are said to be supplementary if the sum of their measures is $180^\circ$.

The supplement of the angle with measure $\theta$ is $180^\circ - \theta$.

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According to the problem statement, the angle is equal to its supplement.

Therefore, we can write the equation:

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Now, we need to solve this equation for $\theta$.

Add $\theta$ to both sides of the equation:

---

Divide both sides by 2:

---

So, the measure of the angle is $90^\circ$.

Let's verify the answer. If the angle is $90^\circ$, its supplement is $180^\circ - 90^\circ = 90^\circ$. Since $90^\circ = 90^\circ$, the angle is indeed equal to its supplement.

---

Looking at the options:

(A) $45^\circ$

(B) $90^\circ$

(C) $180^\circ$

(D) $0^\circ$

The calculated measure $90^\circ$ matches option (B).

---

The correct answer is (B) $90^\circ$.

The question states that $\angle A$ and $\angle B$ form a linear pair and provides the measure of $\angle A$. We need to find the measure of $\angle B$.

---

A linear pair of angles is a pair of adjacent angles formed when two lines intersect. The sum of the measures of the angles in a linear pair is always $180^\circ$. This is known as the Linear Pair Axiom.

---

Given that $\angle A$ and $\angle B$ form a linear pair, we have:

---

We are given that $\angle A = 85^\circ$. Substituting this value into the equation:

---

To find the measure of $\angle B$, subtract $85^\circ$ from both sides of the equation:

---

Thus, the measure of angle B is $95^\circ$.

---

Comparing the result with the given options:

(A) $5^\circ$

(B) $95^\circ$

(C) $85^\circ$

(D) $105^\circ$

The calculated value $\angle B = 95^\circ$ matches option (B).

---

The correct answer is (B) $95^\circ$.

The question asks to identify the true statement about two parallel lines.

---

Let's consider the definition of parallel lines in geometry.

Two distinct lines are said to be parallel if they lie in the same plane and do not intersect, no matter how far they are extended.

---

Now, let's examine each of the given options based on this definition.

---

(A) They meet at one point.

This statement describes intersecting lines, not parallel lines. Intersecting lines cross each other at exactly one point. Therefore, this statement is false for parallel lines.

---

(B) They diverge from each other.

Parallel lines maintain a constant distance between them. They do not get further apart (diverge) or closer together. Lines that diverge might refer to lines that are not in the same plane (skew lines) or possibly lines that are not parallel and are moving away from each other after intersection, but it's not the definition of parallel lines. Therefore, this statement is generally false in the context of parallel lines in a plane.

---

(C) They lie in the same plane and do not intersect.

This statement is the precise definition of parallel lines in Euclidean geometry. They are coplanar (lie in the same plane) and non-intersecting. Therefore, this statement is true.

---

(D) They are always vertical.

Parallel lines can have any orientation as long as they are in the same plane and have the same slope (in coordinate geometry). For example, two horizontal lines are parallel to each other, two vertical lines are parallel to each other, and two lines with a slope of 1 are parallel to each other. They are not restricted to being vertical. Therefore, this statement is false.

---

Based on the analysis of the options, the only statement that is true if two lines are parallel is that they lie in the same plane and do not intersect.

---

The correct answer is (C) They lie in the same plane and do not intersect.

The question describes a scenario where a transversal intersects two lines, and the sum of the interior angles on the same side of the transversal is $180^\circ$. We need to determine the relationship between the two lines based on this condition.

---

When a transversal intersects two lines, it forms several pairs of angles. The interior angles on the same side of the transversal are also known as consecutive interior angles or co-interior angles.

---

Let the two lines be $l$ and $m$, and let the transversal be $t$. Let the interior angles on the same side of the transversal be $\angle 1$ and $\angle 2$. The given condition is:

---

There is a fundamental property relating parallel lines and consecutive interior angles: If two parallel lines are intersected by a transversal, then the sum of the consecutive interior angles on the same side of the transversal is $180^\circ$.

---

The converse of this property is also true and is used as a test for parallelism: If a transversal intersects two lines such that the sum of the interior angles on the same side of the transversal is $180^\circ$, then the two lines are parallel.

---

The given condition exactly matches the condition in the converse property. Therefore, based on this property, if the sum of the interior angles on the same side is $180^\circ$, the lines must be parallel.

---

Let's look at the given options:

(A) Perpendicular: Perpendicular lines intersect at $90^\circ$. This is not implied by the condition.

(B) Intersecting: The lines are intersected by a transversal, but the condition forces a specific relationship (parallelism) which prevents intersection.

(C) Parallel: This matches our conclusion based on the converse of the consecutive interior angles property.

(D) Skew: Skew lines are non-parallel and non-intersecting lines in three-dimensional space. The problem is set in a context where angles and transversals are discussed, implying a planar geometry setting, or at least the lines are coplanar with the transversal where the angles are formed. Thus, skew lines are not relevant here.

---

The condition that the sum of interior angles on the same side is $180^\circ$ is a sufficient condition for the two lines to be parallel.

---

The correct answer is (C) Parallel.

The question asks for the complement of an angle $x$.

---

In geometry, two angles are said to be complementary if the sum of their measures is $90^\circ$.

---

Let the given angle be $x$.

Let its complement be $y$.

---

By the definition of complementary angles, the sum of the angle and its complement is $90^\circ$.

---

To find the complement ($y$), we need to isolate $y$ in the equation. Subtract $x$ from both sides:

---

So, the complement of an angle $x$ is $90^\circ - x$.

---

Now, let's compare this result with the given options:

(A) $x + 90^\circ$

(B) $90^\circ - x$

(C) $180^\circ - x$

(D) $x - 90^\circ$

Our result, $90^\circ - x$, matches option (B).

Note that $180^\circ - x$ represents the supplement of the angle $x$, not the complement.

---

The correct answer is (B) $90^\circ - x$.

The question asks to identify which of the given pairs of angles are adjacent angles.

---

Let's first recall the definition of adjacent angles.

Two angles are called adjacent angles if they have a common vertex and a common arm (side), and their non-common arms are on opposite sides of the common arm.

---

Now let's examine each option:

(A) Two vertically opposite angles

Vertically opposite angles are formed by the intersection of two lines. They share a common vertex, but their arms are opposite rays forming two pairs of opposite arms. They do not share a common arm. Therefore, vertically opposite angles are not adjacent angles.

---

(B) Two angles that form a linear pair

A linear pair consists of two adjacent angles whose non-common arms are opposite rays (forming a straight line). By definition, angles forming a linear pair are always adjacent angles. They share a common vertex and a common arm, and their non-common arms lie on a straight line (which means they are opposite rays). This fits the definition of adjacent angles.

---

(C) Two alternate interior angles

Alternate interior angles are formed when a transversal line intersects two other lines. They are located on opposite sides of the transversal and between the two lines. They do not share a common vertex or a common arm. Therefore, alternate interior angles are not adjacent angles.

---

(D) Two complementary angles that do not share a vertex or arm

This option explicitly states that the two complementary angles do not share a vertex or arm. By the definition of adjacent angles, they must share a common vertex and a common arm. Therefore, these angles are not adjacent angles. Complementary angles can be adjacent (e.g., two angles that form a right angle when placed side by side), but they are not always adjacent.

---

Based on the definitions, the only pair of angles among the options that are always adjacent angles is a linear pair.

---

The correct answer is (B) Two angles that form a linear pair.

The question asks which statement is always true if $\angle 1$ and $\angle 2$ are complementary.

---

Let's recall the definition of complementary angles.

Two angles are said to be complementary if the sum of their measures is $90^\circ$.

---

Given that $\angle 1$ and $\angle 2$ are complementary, by the definition:

---

Now, let's examine the given options:

(A) $\angle 1 = \angle 2$

This statement is true only if $\angle 1$ and $\angle 2$ are both equal to $45^\circ$ ($45^\circ + 45^\circ = 90^\circ$). However, complementary angles do not have to be equal. For example, angles measuring $30^\circ$ and $60^\circ$ are complementary ($30^\circ + 60^\circ = 90^\circ$), but they are not equal. Thus, this statement is not always true.

---

(B) $\angle 1 + \angle 2 = 90^\circ$

This statement is the very definition of complementary angles. Therefore, if $\angle 1$ and $\angle 2$ are complementary, this statement is always true.

---

(C) $\angle 1 + \angle 2 = 180^\circ$

This statement describes supplementary angles, not complementary angles. Thus, this statement is not always true for complementary angles.

---

(D) $\angle 1$ and $\angle 2$ are adjacent

Complementary angles can be adjacent (e.g., two angles forming a right angle side by side), but they do not necessarily have to be adjacent. For example, an angle in one part of a diagram and another angle in a completely different part of the diagram can be complementary without sharing a vertex or an arm. Thus, this statement is not always true.

---

Based on the definition and the analysis of the options, the only statement that is always true when $\angle 1$ and $\angle 2$ are complementary is that their sum is $90^\circ$.

---

The correct answer is (B) $\angle 1 + \angle 2 = 90^\circ$.

The question asks to complete the statement describing a specific pair of angles formed when a transversal intersects two lines.

---

The statement describes angles that are on the same side of a transversal and are located inside the two lines.

---

Let's consider the types of angles formed by a transversal intersecting two lines:

• Interior Angles: These are the angles that lie between the two lines.
• Exterior Angles: These are the angles that lie outside the two lines.
• Alternate Interior Angles: Pairs of interior angles on opposite sides of the transversal.
• Alternate Exterior Angles: Pairs of exterior angles on opposite sides of the transversal.
• Corresponding Angles: Pairs of angles that are in the same relative position at each intersection (e.g., top-left, bottom-right), on the same side of the transversal. One is interior and one is exterior.
• Consecutive Interior Angles (or Co-interior Angles): Pairs of interior angles that are on the same side of the transversal.

---

The description "Angles on the same side of a transversal and inside the two lines" precisely matches the definition of Consecutive Interior Angles.

---

Now let's check the given options:

(A) Alternate: Alternate interior angles are on opposite sides of the transversal.

(B) Corresponding: Corresponding angles are on the same side, but one is interior and the other is exterior.

(C) Consecutive: Consecutive interior angles are on the same side of the transversal and are interior.

(D) Vertically opposite: Vertically opposite angles are formed at a single intersection and are opposite to each other.

---

The term that correctly completes the statement is "Consecutive".

---

The correct answer is (C) Consecutive.

The question states that a transversal intersects two parallel lines and that one of the angles formed has a measure of $40^\circ$. We need to determine which of the given measures CANNOT be the measure of another angle formed.

---

When a transversal intersects two parallel lines, the eight angles formed have specific relationships:

• Angles forming a linear pair are supplementary (sum to $180^\circ$).
• Vertically opposite angles are equal.
• Corresponding angles are equal.
• Alternate interior angles are equal.
• Alternate exterior angles are equal.
• Consecutive interior (or co-interior) angles are supplementary (sum to $180^\circ$).

---

Given that one angle is $40^\circ$. Let's denote this angle as $\alpha = 40^\circ$.

Based on the properties of angles formed by a transversal intersecting parallel lines, the measure of any other angle formed must be either equal to $\alpha$ or supplementary to $\alpha$.

The supplement of $\alpha$ is $180^\circ - \alpha = 180^\circ - 40^\circ = 140^\circ$.

Therefore, the only possible measures for any other angle formed are $40^\circ$ or $140^\circ$.

---

Let's check the given options:

(A) $40^\circ$

An angle of $40^\circ$ is possible (e.g., vertically opposite to the given angle, corresponding angle, alternate angle).

---

(B) $140^\circ$

An angle of $140^\circ$ is possible (e.g., forming a linear pair with the given $40^\circ$ angle, or a consecutive interior angle with the given $40^\circ$ angle if it is an interior angle). The supplement of $40^\circ$ is $140^\circ$.

---

(C) $50^\circ$

An angle of $50^\circ$ is neither $40^\circ$ nor $140^\circ$. Therefore, it cannot be the measure of any angle formed when a transversal intersects two parallel lines and one angle is $40^\circ$.

---

(D) The vertically opposite angle to $40^\circ$

The vertically opposite angle to an angle of $40^\circ$ is always equal to $40^\circ$. As established, $40^\circ$ is a possible angle measure.

---

The only measure among the options that cannot be formed is $50^\circ$.

---

The correct answer is (C) $50^\circ$.

The question states that two angles are supplementary and that one angle is twice the other. We need to find the measure of the larger angle.

---

Let the measures of the two angles be $x$ and $y$.

We are given that the two angles are supplementary. This means their sum is $180^\circ$.

---

We are also given that one angle is twice the other. Let's assume the first angle $x$ is twice the second angle $y$.

---

Now we have a system of two linear equations with two variables. We can solve this system to find the values of $x$ and $y$.

Substitute equation (ii) into equation (i):

---

Now, solve for $y$ by dividing both sides by 3:

---

Now that we have the value of $y$, we can find the value of $x$ using equation (ii):

---

The two angles are $60^\circ$ and $120^\circ$.

Check: Are they supplementary? $60^\circ + 120^\circ = 180^\circ$. Yes.

Check: Is one angle twice the other? $120^\circ = 2 \times 60^\circ$. Yes.

---

The question asks for the measure of the larger angle. Comparing the two angles, $120^\circ$ is larger than $60^\circ$.

---

Looking at the given options:

(A) $60^\circ$

(B) $120^\circ$

(C) $90^\circ$

(D) $30^\circ$

The larger angle is $120^\circ$, which matches option (B).

---

The correct answer is (B) $120^\circ$.

The question asks to find the angle that would form a linear pair with an angle of $72^\circ$.

---

A linear pair of angles is a pair of adjacent angles formed when two lines intersect. The sum of the measures of the angles in a linear pair is always $180^\circ$.

---

Let the given angle be $\angle A$ with measure $72^\circ$.

Let the angle that forms a linear pair with $\angle A$ be $\angle B$.

---

Since $\angle A$ and $\angle B$ form a linear pair, their sum must be $180^\circ$.

---

Substitute the value of $\angle A$ into the equation:

---

To find the measure of $\angle B$, subtract $72^\circ$ from both sides of the equation:

---

Thus, the angle that forms a linear pair with an angle of $72^\circ$ has a measure of $108^\circ$.

---

Let's consider the options provided:

(A) Its complement ($18^\circ$)

The complement of $72^\circ$ is $90^\circ - 72^\circ = 18^\circ$. An angle of $18^\circ$ does not form a linear pair with $72^\circ$ as $72^\circ + 18^\circ = 90^\circ \neq 180^\circ$.

(B) Its supplement ($108^\circ$)

The supplement of $72^\circ$ is $180^\circ - 72^\circ = 108^\circ$. An angle of $108^\circ$ forms a linear pair with $72^\circ$ as $72^\circ + 108^\circ = 180^\circ$.

(C) Itself ($72^\circ$)

$72^\circ$ does not form a linear pair with itself as $72^\circ + 72^\circ = 144^\circ \neq 180^\circ$.

(D) Its vertically opposite angle ($72^\circ$)

The vertically opposite angle to $72^\circ$ is $72^\circ$. Vertically opposite angles do not form a linear pair unless they are both $90^\circ$. Also, a vertically opposite angle to $72^\circ$ is equal to $72^\circ$, not $108^\circ$.

---

The angle that forms a linear pair with $72^\circ$ is its supplement, which is $108^\circ$. This matches option (B).

---

The correct answer is (B) Its supplement ($108^\circ$).

The question describes three lines, A, B, and C, with given relationships between them, and asks for the relationship between line A and line C.

---

Given:

• Line A is parallel to line B ($A \parallel B$).
• Line B is parallel to line C ($B \parallel C$).

---

To Find:

The relationship between line A and line C.

---

Solution:

This problem involves a fundamental property of parallel lines, often referred to as the Transitive Property of Parallel Lines.

---

The Transitive Property of Parallel Lines states that if two distinct lines are each parallel to a third line, then the two lines are parallel to each other. This property applies when all three lines are in the same plane (coplanar), which is typically assumed in such problems unless stated otherwise.

---

In this case, we are given that Line A is parallel to Line B, and Line B is parallel to Line C. Line B is the third line to which both Line A and Line C are parallel.

Applying the Transitive Property:

Since $A \parallel B$ and $B \parallel C$, it follows that $A \parallel C$.

---

Therefore, line A is parallel to line C.

---

Now, let's compare this result with the given options:

(A) Perpendicular: If A were perpendicular to C, and A is parallel to B, then B would also be perpendicular to C. However, we are given B is parallel to C, which contradicts them being perpendicular (unless they are the same line, but distinct lines are usually implied). This option is incorrect.

(B) Intersecting: If A and C were intersecting, it would contradict the property that if A is parallel to B and B is parallel to C, then A is parallel to C. Parallel lines do not intersect. This option is incorrect.

(C) Parallel: This matches our conclusion based on the Transitive Property of Parallel Lines.

(D) Transversal: A transversal is a line that intersects two or more other lines. While A could be a transversal to some other lines, the relationship between A and C is one of parallelism under the given conditions. This option describes a role a line can play, not the intrinsic relationship between A and C here. This option is incorrect.

---

The statement that is true about line A and line C is that they are parallel.

---

The correct answer is (C) Parallel.

The question describes two intersecting lines and provides the measure of one of the angles formed. We need to find the measures of the other three angles.

---

When two lines intersect, they form four angles at the point of intersection. Let's denote the angles as $\angle 1, \angle 2, \angle 3,$ and $\angle 4$ in a circular order around the intersection point. Suppose the given angle is $\angle 1 = 55^\circ$.

---

Two pairs of vertically opposite angles are formed by the intersection of two lines. Vertically opposite angles are equal in measure.

In this case, $\angle 1$ is vertically opposite to $\angle 3$, and $\angle 2$ is vertically opposite to $\angle 4$.

Since $\angle 1 = 55^\circ$, its vertically opposite angle $\angle 3$ must also be $55^\circ$.

---

Also, adjacent angles formed by intersecting lines constitute linear pairs. Angles in a linear pair are supplementary, meaning their sum is $180^\circ$.

$\angle 1$ and $\angle 2$ form a linear pair. Therefore:

---

Substitute the value of $\angle 1 = 55^\circ$ into equation (i):

Subtract $55^\circ$ from both sides to find $\angle 2$:

---

Similarly, $\angle 1$ and $\angle 4$ form a linear pair. Therefore, $\angle 4 = 180^\circ - \angle 1 = 180^\circ - 55^\circ = 125^\circ$. Alternatively, $\angle 4$ is vertically opposite to $\angle 2$, so $\angle 4 = \angle 2 = 125^\circ$.

---

The measures of the other three angles are $\angle 2 = 125^\circ$, $\angle 3 = 55^\circ$, and $\angle 4 = 125^\circ$.

So the measures are $55^\circ, 125^\circ, 125^\circ$ (in any order).

---

Comparing this result with the given options:

(A) $55^\circ, 125^\circ, 125^\circ$

(B) $55^\circ, 55^\circ, 125^\circ$

(C) $125^\circ, 125^\circ, 125^\circ$

(D) $55^\circ, 35^\circ, 35^\circ$

The calculated measures match option (A).

---

The correct answer is (A) $55^\circ, 125^\circ, 125^\circ$.

The question asks about the property that ensures railway tracks remain parallel.

---

Railway tracks are designed to be parallel lines. This is crucial for the safe and smooth movement of trains, as the wheels on each side of the train maintain a fixed distance apart.

---

Let's consider the properties related to parallel lines and the given options.

• Parallel lines are lines in the same plane that never intersect.
• A key characteristic of parallel lines is that the perpendicular distance between them is the same at all points.

---

Now let's examine the options:

(A) Vertically opposite angles are equal.

This property is true for any pair of intersecting lines. It is not a property that specifically defines or ensures parallelism between two separate lines like railway tracks.

---

(B) Angles in a linear pair are supplementary.

This property is true for any two adjacent angles that form a straight line. It is not a property that defines or ensures parallelism between two separate lines.

---

(C) The distance between them is constant.

This is a fundamental property of parallel lines. If the distance between two lines in a plane is constant, they are parallel, and conversely, if two lines are parallel, the perpendicular distance between them is constant. This is the property that is directly relevant to how railway tracks are constructed and maintained to ensure they stay parallel.

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(D) Corresponding angles are equal when intersected by a perpendicular.

When parallel lines are intersected by *any* transversal, corresponding angles are equal. This statement is a specific instance of that property (when the transversal is perpendicular), but the equality of corresponding angles is a consequence of parallelism, not the property that ensures parallelism in the context of construction like railway tracks. The constant distance is the more direct and practical property in this context.

---

The property that directly ensures that railway tracks remain parallel, from a construction and geometric perspective, is that the distance between them is kept constant along their entire length.

---

The correct answer is (C) The distance between them is constant.

The question asks to find the measure of an angle that is $30^\circ$ more than its complement.

---

Given:

An angle is $30^\circ$ more than its complement.

---

To Find:

The measure of the angle.

---

Solution:

Let the measure of the angle be $x$ degrees.

The complement of an angle $x$ is the angle that, when added to $x$, results in $90^\circ$. The measure of the complement of angle $x$ is $90^\circ - x$.

---

According to the problem statement, the angle ($x$) is $30^\circ$ more than its complement ($90^\circ - x$).

We can write this relationship as an equation:

---

Now, we need to solve this equation for $x$.

Simplify the right side of the equation:

---

To isolate $x$ on one side, add $x$ to both sides of the equation:

---

Now, solve for $x$ by dividing both sides by 2:

---

The measure of the angle is $60^\circ$.

---

Let's verify the answer. If the angle is $60^\circ$, its complement is $90^\circ - 60^\circ = 30^\circ$. Is the angle $30^\circ$ more than its complement? Yes, $60^\circ = 30^\circ + 30^\circ$. The condition is satisfied.

---

Comparing the result with the given options:

(A) $30^\circ$

(B) $60^\circ$

(C) $45^\circ$

(D) $75^\circ$

The calculated value $x = 60^\circ$ matches option (B).

---

The correct answer is (B) $60^\circ$.

The question asks to identify which of the given pairs of angles are formed when a transversal intersects two lines.

---

When a transversal line intersects two other lines, it creates eight angles at the two points of intersection. These angles have specific names and relationships, especially when the two lines are parallel.

---

Let's consider the types of angle pairs mentioned in the options:

(A) Corresponding angles

Corresponding angles are indeed formed when a transversal intersects two lines. They are pairs of angles that occupy the same relative position at each intersection where the transversal crosses the other lines. For example, the top-left angle at the first intersection and the top-left angle at the second intersection are corresponding angles.

---

(B) Alternate interior angles

Alternate interior angles are also formed when a transversal intersects two lines. They are pairs of interior angles on opposite sides of the transversal. They lie between the two lines but on alternate sides of the transversal.

---

(C) Vertically opposite angles (at each intersection)

At each point where the transversal intersects one of the lines, two pairs of vertically opposite angles are formed. Since there are two intersection points, a transversal intersecting two lines forms a total of two pairs of vertically opposite angles, one pair at each intersection.

---

Since corresponding angles, alternate interior angles, and vertically opposite angles are all types of angle pairs formed when a transversal intersects two lines, the statement "All of the above" is correct.

---

The correct answer is (D) All of the above.

The question describes two lines $p$ and $q$ intersected by a transversal $r$, and states that the alternate interior angles formed are not equal. We need to determine the relationship between lines $p$ and $q$ based on this condition.

---

Let's recall the property concerning alternate interior angles and parallel lines. When a transversal intersects two lines, the alternate interior angles are the pairs of interior angles on opposite sides of the transversal.

---

The fundamental property related to this is: If two parallel lines are intersected by a transversal, then the alternate interior angles are equal.

---

The converse of this property is also a fundamental theorem used to determine if lines are parallel: If a transversal intersects two lines such that the alternate interior angles are equal, then the two lines are parallel.

---

The question gives us the condition that the alternate interior angles are not equal. This is the negation of the condition in the converse statement. Therefore, based on the converse property, if the alternate interior angles are not equal, then the two lines cannot be parallel.

---

So, we conclude that lines $p$ and $q$ are not parallel.

---

In Euclidean geometry, two distinct lines in the same plane can either be parallel or intersect at exactly one point. Since lines $p$ and $q$ are not parallel (and assuming they are in the same plane and are distinct lines, which is implied by the terms 'lines $p$ and $q$' being 'intersected by a transversal $r$'), they must intersect.

---

Let's look at the given options:

(A) Parallel: This contradicts our conclusion.

(B) Perpendicular: Perpendicular lines are a specific type of intersecting lines, where the angles of intersection are $90^\circ$. While the lines are intersecting, they are not necessarily perpendicular just because the alternate interior angles are not equal. They could intersect at any angle other than one which would make alternate interior angles equal.

(C) Intersecting: This aligns with our conclusion that if the lines are not parallel (and are coplanar and distinct), they must intersect.

(D) Coincident: Coincident lines are the same line. If $p$ and $q$ were the same line, any transversal would form equal alternate interior angles, contradicting the given condition.

---

Therefore, if the alternate interior angles formed by a transversal intersecting two lines $p$ and $q$ are not equal, the lines $p$ and $q$ must be intersecting.

---

The correct answer is (C) Intersecting.

The question asks to identify the supplementary angle to an angle that measures $100^\circ$.

---

Let the given angle be $\angle A$ with measure $100^\circ$.

The supplementary angle to $\angle A$ is an angle that, when added to $\angle A$, results in a sum of $180^\circ$. Let the supplementary angle be $\angle B$.

---

Substitute the measure of $\angle A$:

---

Subtract $100^\circ$ from both sides to find the measure of $\angle B$:

---

So, the supplementary angle to an angle measuring $100^\circ$ is an angle measuring $80^\circ$.

---

Now, let's examine the given options:

(A) An angle that forms a linear pair with it.

Angles in a linear pair are supplementary. If an angle measures $100^\circ$, the angle that forms a linear pair with it will have a measure of $180^\circ - 100^\circ = 80^\circ$. This angle is indeed supplementary to the given angle.

---

(B) An angle measuring $80^\circ$.

As calculated above, the supplementary angle to $100^\circ$ has a measure of $80^\circ$. This directly matches the measure of the supplementary angle.

---

(C) An angle whose sum with the given angle is $180^\circ$.

By the definition of supplementary angles, two angles are supplementary if their sum is $180^\circ$. So, an angle whose sum with $100^\circ$ is $180^\circ$ is precisely the supplementary angle. This angle would measure $180^\circ - 100^\circ = 80^\circ$.

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Option (A) describes a relationship (linear pair) that results in supplementary angles. Option (B) gives the specific measure of the supplementary angle. Option (C) restates the definition of supplementary angles in terms of the given angle. All three statements correctly describe or identify the supplementary angle to a $100^\circ$ angle.

---

Since options (A), (B), and (C) are all correct descriptions related to the supplementary angle of $100^\circ$, the correct answer is (D).

---

The correct answer is (D) All of the above.

What are complementary angles?

---

Two angles are called complementary angles if the sum of their measures is $90^\circ$.

---

Finding the measure of the other angle:

Let the measure of the two complementary angles be $\theta_1$ and $\theta_2$.

By the definition of complementary angles, we have:

$\theta_1 + \theta_2 = 90^\circ$

---

We are given that one of the angles is $48^\circ$. Let's assume $\theta_1 = 48^\circ$.

We need to find the measure of the other angle, $\theta_2$.

---

Substitute the given value into the equation:

$48^\circ + \theta_2 = 90^\circ$

---

To find $\theta_2$, subtract $48^\circ$ from both sides of the equation:

$\theta_2 = 90^\circ - 48^\circ$

$\theta_2 = 42^\circ$

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Thus, the measure of the other angle is $42^\circ$.

---

We can verify this: $48^\circ + 42^\circ = 90^\circ$, so they are indeed complementary.

Define supplementary angles:

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Two angles are called supplementary angles if the sum of their measures is $180^\circ$.

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Finding the measure of its supplementary angle:

Let the given angle be $\angle A$ with measure $115^\circ$.

Let the measure of its supplementary angle be $\angle B$.

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By the definition of supplementary angles, the sum of the two angles is $180^\circ$.

---

Substitute the given measure of $\angle A$ into the equation:

---

To find the measure of $\angle B$, subtract $115^\circ$ from both sides of the equation:

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Thus, the measure of the supplementary angle is $65^\circ$.

---

We can verify this: $115^\circ + 65^\circ = 180^\circ$, so they are indeed supplementary.

No, two obtuse angles cannot be supplementary.

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Reason:

An obtuse angle is an angle whose measure is greater than $90^\circ$ but less than $180^\circ$.

Let the measures of two obtuse angles be $\alpha$ and $\beta$.

By the definition of an obtuse angle:

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For two angles to be supplementary, the sum of their measures must be exactly $180^\circ$.

---

Now, let's consider the sum of the two obtuse angles $\alpha$ and $\beta$. Since $\alpha > 90^\circ$ and $\beta > 90^\circ$, their sum will be:

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The sum of two obtuse angles is always greater than $180^\circ$. Therefore, their sum cannot be equal to $180^\circ$.

---

Hence, two obtuse angles cannot be supplementary.

Define adjacent angles:

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Two angles are called adjacent angles if they have a common vertex, a common arm (or side), and their non-common arms are on opposite sides of the common arm.

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Diagram showing two adjacent angles $\angle AOC$ and $\angle COB$ such that they form a linear pair:

A linear pair consists of two adjacent angles whose non-common arms are opposite rays, forming a straight line.

Consider a straight line AB with a point O on it.

Draw a ray OC originating from O such that it is not along the line AB.

The angles formed are $\angle AOC$ and $\angle COB$.

These angles share the common vertex O and the common arm OC.

The non-common arms are the rays OA and OB, which are opposite rays since they lie on the straight line AB.

Since they satisfy the conditions of adjacent angles and their non-common arms form a straight line, they form a linear pair.

(Diagram not displayable in this format, but should represent a straight line AB, with point O on it, and a ray OC emanating from O).

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In the diagram:

• Common vertex: O
• Common arm: Ray OC
• Non-common arms: Rays OA and OB
• Rays OA and OB are opposite rays (lie on the straight line AB)

Therefore, $\angle AOC$ and $\angle COB$ are adjacent angles that form a linear pair.

A linear pair of angles is a pair of adjacent angles formed when two lines intersect. The angles share a common vertex and a common side, and their non-common sides are opposite rays.

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The sum of the measures of angles forming a linear pair is always $180^\circ$.

Given:

Two angles form a linear pair.

Measure of one angle is $70^\circ$.

---

To Find:

The measure of the other angle.

---

Solution:

Let the measure of the other angle be $x$.

Angles forming a linear pair are supplementary, which means their sum is $180^\circ$.

So, we have the equation:

Subtract $70^\circ$ from both sides of the equation:

$x = 180^\circ - 70^\circ$

$x = 110^\circ$

---

Therefore, the measure of the other angle is $110^\circ$.

Vertically opposite angles are the pairs of angles formed when two lines intersect. They are opposite to each other at the point of intersection. A key property of vertically opposite angles is that they are equal in measure.

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Drawing and Labelling:

1. Draw two straight lines that intersect each other at a point. Let the lines be $AB$ and $CD$, and let the point of intersection be $O$.

2. The angles formed are $\angle AOC$, $\angle COB$, $\angle BOD$, and $\angle DOA$.

3. There are two pairs of vertically opposite angles:

- $\angle AOC$ and $\angle BOD$ are a pair of vertically opposite angles.

- $\angle COB$ and $\angle DOA$ are another pair of vertically opposite angles.

4. To label, you can mark $\angle AOC$ and $\angle BOD$ with the same arc or symbol to indicate they are vertically opposite and equal.

Given:

Two lines intersect.

Measure of one angle formed is $40^\circ$.

---

To Find:

The measure of the angle vertically opposite to it.

---

Solution:

Vertically opposite angles are equal in measure.

Since one angle is $40^\circ$, the measure of the angle vertically opposite to it is also $40^\circ$.

---

Therefore, the measure of the angle vertically opposite to the $40^\circ$ angle is $40^\circ$.

To Find:

The measure of an angle that is equal to its complement.

---

Solution:

Let the measure of the angle be $x$.

The measure of its complement is $90^\circ - x$, because complementary angles add up to $90^\circ$.

According to the problem, the angle is equal to its complement.

So, we can write the equation:

$x = 90^\circ - x$

Add $x$ to both sides of the equation:

$x + x = 90^\circ$

$2x = 90^\circ$

Divide both sides by 2:

$x = \frac{90^\circ}{2}$

$x = 45^\circ$

---

The measure of the angle which is equal to its complement is $45^\circ$.

Check: The complement of $45^\circ$ is $90^\circ - 45^\circ = 45^\circ$, which is equal to the angle itself.

To Find:

The measure of an angle that is equal to its supplement.

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Solution:

Let the measure of the angle be $x$.

The measure of its supplement is $180^\circ - x$, because supplementary angles add up to $180^\circ$.

According to the problem, the angle is equal to its supplement.

So, we can write the equation:

$x = 180^\circ - x$

Add $x$ to both sides of the equation:

$x + x = 180^\circ$

$2x = 180^\circ$

Divide both sides by 2:

$x = \frac{180^\circ}{2}$

$x = 90^\circ$

---

The measure of the angle which is equal to its supplement is $90^\circ$.

Check: The supplement of $90^\circ$ is $180^\circ - 90^\circ = 90^\circ$, which is equal to the angle itself.

Given:

Two angles are in the ratio $2:3$.

The two angles are complementary.

---

To Find:

The measures of the two angles.

---

Solution:

Let the measures of the two angles be $2x$ and $3x$, where $x$ is a common factor.

Since the angles are complementary, their sum is $90^\circ$.

So, we have the equation:

$2x + 3x = 90^\circ$

Combine the terms on the left side:

$5x = 90^\circ$

Divide both sides by 5 to find the value of $x$:

$x = \frac{90^\circ}{5}$

$x = 18^\circ$

Now, we can find the measures of the two angles:

First angle = $2x = 2 \times 18^\circ = 36^\circ$

Second angle = $3x = 3 \times 18^\circ = 54^\circ$

---

The measures of the two angles are $36^\circ$ and $54^\circ$.

Check: $36^\circ + 54^\circ = 90^\circ$, so they are complementary. The ratio $36:54$ simplifies to $2:3$ (dividing both by 18).

Given:

Two angles are in the ratio $4:5$.

The two angles are supplementary.

---

To Find:

The measures of the two angles.

---

Solution:

Let the measures of the two angles be $4x$ and $5x$, where $x$ is a common factor.

Since the angles are supplementary, their sum is $180^\circ$.

So, we have the equation:

$4x + 5x = 180^\circ$

Combine the terms on the left side:

$9x = 180^\circ$

Divide both sides by 9 to find the value of $x$:

$x = \frac{180^\circ}{9}$

$x = 20^\circ$

Now, we can find the measures of the two angles:

First angle = $4x = 4 \times 20^\circ = 80^\circ$

Second angle = $5x = 5 \times 20^\circ = 100^\circ$

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The measures of the two angles are $80^\circ$ and $100^\circ$.

Check: $80^\circ + 100^\circ = 180^\circ$, so they are supplementary. The ratio $80:100$ simplifies to $4:5$ (dividing both by 20).

A transversal line is a line that intersects two or more distinct lines at different points.

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Diagram showing a transversal intersecting two lines:

1. Draw two lines, say line $l_1$ and line $l_2$. These lines can be parallel or not parallel.

2. Draw a third line, say line $t$, that crosses both line $l_1$ and line $l_2$. Ensure that line $t$ intersects $l_1$ and $l_2$ at two different points.

3. Line $t$ is the transversal.

Example Diagram Description:

Imagine a horizontal line $l_1$ and another horizontal line $l_2$ below it. Now, imagine a line $t$ crossing both of these lines diagonally from top-left to bottom-right.

The intersection points could be labeled, for example, $P$ where $t$ intersects $l_1$, and $Q$ where $t$ intersects $l_2$.

Corresponding angles are pairs of angles that are in the same relative position at each intersection where a transversal line intersects two or more lines.

---

Assuming the angles are labelled 1 to 8 in a standard way (e.g., angles 1, 2, 3, 4 at the first intersection and angles 5, 6, 7, 8 at the second intersection, following a consistent order like clockwise from the top-left):

---

A pair of corresponding angles is $\angle 1$ and $\angle 5$.

---

Other possible pairs of corresponding angles are:

- $\angle 2$ and $\angle 6$

- $\angle 4$ and $\angle 8$

- $\angle 3$ and $\angle 7$

Alternate interior angles are pairs of angles that are on opposite sides of the transversal and lie between the two lines that the transversal intersects.

---

Assuming the standard labelling from Question 14 (angles 1, 2, 3, 4 at the first intersection, 5, 6, 7, 8 at the second, where 3, 4, 5, 6 are interior angles):

---

A pair of alternate interior angles is $\angle 4$ and $\angle 5$.

---

Another pair of alternate interior angles is $\angle 3$ and $\angle 6$.

Interior angles on the same side of the transversal (also known as consecutive interior angles or same-side interior angles) are pairs of angles that are on the same side of the transversal and lie between the two lines that the transversal intersects.

---

Assuming the standard labelling from Question 14 (angles 1, 2, 3, 4 at the first intersection, 5, 6, 7, 8 at the second, where 3, 4, 5, 6 are interior angles):

---

A pair of interior angles on the same side of the transversal is $\angle 4$ and $\angle 6$.

---

Another pair of interior angles on the same side of the transversal is $\angle 3$ and $\angle 5$.

When two parallel lines are intersected by a transversal, each pair of corresponding angles is equal in measure.

When two parallel lines are intersected by a transversal, each pair of alternate interior angles is equal in measure.

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Explanation:

Consider two parallel lines, $l_1$ and $l_2$, intersected by a transversal $t$. Let's label a pair of alternate interior angles as $\angle 1$ and $\angle 2$ (where $\angle 1$ is on $l_1$, $\angle 2$ is on $l_2$, they are between the lines, and on opposite sides of $t$).

Let's also consider the angle $\angle 3$ which is vertically opposite to $\angle 2$. Vertically opposite angles are always equal, so $\angle 2 = \angle 3$.

Now, observe $\angle 1$ and $\angle 3$. Angle $\angle 1$ (on line $l_1$) and angle $\angle 3$ (on line $l_2$) are corresponding angles because they are in the same relative position at each intersection.

Since the lines $l_1$ and $l_2$ are parallel, the corresponding angles are equal.

We also know that $\angle 2 = \angle 3$ (Vertically opposite angles).

By the transitive property of equality (if $\angle 1 = \angle 3$ and $\angle 2 = \angle 3$), we can conclude that:

Thus, the pair of alternate interior angles $(\angle 1 \text{ and } \angle 2)$ are equal in measure.

This relationship holds true for any pair of alternate interior angles formed when a transversal intersects two parallel lines.

When two parallel lines are intersected by a transversal, the interior angles on the same side of the transversal are supplementary.

---

This means that the sum of the measures of the interior angles on the same side of the transversal is always $180^\circ$.

Given:

Line $l \parallel$ Line $m$

Transversal $t$ intersects $l$ and $m$.

$\angle 1 = 75^\circ$

---

To Find:

The measure of $\angle 5$.

---

Solution:

In the given figure, $\angle 1$ and $\angle 5$ are a pair of corresponding angles.

When a transversal intersects two parallel lines, the corresponding angles are equal in measure.

Since $\angle 1 = 75^\circ$, we have:

$\angle 5 = 75^\circ$

---

Therefore, the measure of $\angle 5$ is $75^\circ$.

Given:

Line $l \parallel$ Line $m$

Transversal $t$ intersects $l$ and $m$.

$\angle 1 = 75^\circ$

---

To Find:

The measure of $\angle 3.

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Solution:

In the given figure, $\angle 1$ and $\angle 3$ are adjacent angles that form a linear pair on line $l$.

Angles forming a linear pair are supplementary, meaning their sum is $180^\circ$.

So, we have the equation:

Substitute the given value of $\angle 1$:

Subtract $75^\circ$ from both sides of the equation:

$\angle 3 = 180^\circ - 75^\circ$

$\angle 3 = 105^\circ$

---

Therefore, the measure of $\angle 3$ is $105^\circ$.

Given:

Line $l \parallel$ Line $m$

Transversal $t$ intersects $l$ and $m$.

$\angle 1 = 75^\circ$

---

To Find:

The measure of $\angle 8$.

---

Solution:

In the given figure, $\angle 1$ and $\angle 5$ are a pair of corresponding angles.

When a transversal intersects two parallel lines, the corresponding angles are equal in measure.

Since $\angle 1 = 75^\circ$, we have:

$\angle 5 = 75^\circ$

Now, consider $\angle 5$ and $\angle 8$. These angles form a linear pair on line $m$.

Angles forming a linear pair are supplementary, meaning their sum is $180^\circ$.

Substitute the value of $\angle 5$:

Subtract $75^\circ$ from both sides of the equation:

$\angle 8 = 180^\circ - 75^\circ$

$\angle 8 = 105^\circ$

---

Therefore, the measure of $\angle 8$ is $105^\circ$.

If a transversal intersects two lines such that the sum of the interior angles on the same side of the transversal is $180^\circ$, then the two lines are parallel.

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This is the converse of the property that states interior angles on the same side of a transversal are supplementary when the lines are parallel.

Yes, two lines that are not parallel can absolutely be intersected by a transversal. When two lines are not parallel, they will intersect at a single point if extended far enough. A transversal is simply a line that intersects two or more other lines at distinct points. So, as long as the third line crosses the two non-parallel lines at two separate points, it is a transversal.

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When a transversal intersects two non-parallel lines, the following types of angles are formed, just like with parallel lines:

1. Corresponding angles

2. Alternate interior angles

3. Alternate exterior angles

4. Interior angles on the same side of the transversal (Consecutive interior angles)

5. Exterior angles on the same side of the transversal (Consecutive exterior angles)

6. Vertically opposite angles

7. Linear pairs

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However, the key difference is that the special relationships between these angle pairs that hold for parallel lines do not hold for non-parallel lines.

- Corresponding angles are not equal.

- Alternate interior angles are not equal.

- Interior angles on the same side of the transversal are not supplementary (their sum is not $180^\circ$).

The relationships that still hold true, regardless of whether the lines are parallel or not, are:

- Vertically opposite angles are always equal.

- Angles forming a linear pair are always supplementary (their sum is $180^\circ$).

(a) Complementary angles:

Two angles are said to be complementary if the sum of their measures is $90^\circ$. Each angle is called the complement of the other.

Illustration:

Imagine a right angle ($90^\circ$). Draw a ray starting from the vertex of the right angle, dividing it into two smaller angles. Let these angles be $\angle A$ and $\angle B$. If $\angle A + \angle B = 90^\circ$, then $\angle A$ and $\angle B$ are complementary angles.

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(b) Supplementary angles:

Two angles are said to be supplementary if the sum of their measures is $180^\circ$. Each angle is called the supplement of the other.

Illustration:

Imagine a straight angle ($180^\circ$), which is formed by a straight line. Draw a ray starting from a point on the straight line (the vertex) and going in any direction not along the line. This ray divides the straight angle into two angles. Let these angles be $\angle C$ and $\angle D$. If $\angle C + \angle D = 180^\circ$, then $\angle C$ and $\angle D$ are supplementary angles.

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(c) Linear pair:

A linear pair is a pair of adjacent angles whose non-common sides are opposite rays (forming a straight line). Angles in a linear pair are always supplementary.

Illustration:

Draw a straight line $AB$ and a point $O$ on it. Draw a ray $OC$ starting from $O$ such that $OC$ is not along the line $AB$. The angles formed are $\angle AOC$ and $\angle BOC$. These angles share a common vertex ($O$) and a common side ($OC$). Their non-common sides ($OA$ and $OB$) are opposite rays forming the straight line $AB$. Thus, $\angle AOC$ and $\angle BOC$ form a linear pair.

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(d) Vertically opposite angles:

Vertically opposite angles are the pairs of angles formed by two intersecting lines. They are opposite to each other at the point of intersection. Vertically opposite angles are always equal in measure.

Illustration:

Draw two straight lines $PQ$ and $RS$ that intersect each other at a point $O$. The angles formed at the intersection are $\angle POR$, $\angle ROQ$, $\angle QOS$, and $\angle SOP$. The pairs of vertically opposite angles are:

- $\angle POR$ and $\angle QOS$

- $\angle ROQ$ and $\angle SOP$

In this illustration, $\angle POR = \angle QOS$ and $\angle ROQ = \angle SOP$.

Given:

Lines AB and CD intersect at O.

$\angle AOC = 55^\circ$

---

To Find:

The measures of $\angle BOD$, $\angle AOD$, and $\angle BOC$.

---

Solution:

Finding $\angle BOD$:

In the given figure, $\angle AOC$ and $\angle BOD$ are vertically opposite angles.

Vertically opposite angles formed by two intersecting lines are always equal in measure.

Since $\angle AOC = 55^\circ$, we have:

$\angle BOD = 55^\circ$

---

Finding $\angle AOD$:

On line CD, the angles $\angle AOC$ and $\angle AOD$ form a linear pair.

Angles forming a linear pair are supplementary, meaning their sum is $180^\circ$.

Substitute the given value of $\angle AOC$:

Subtract $55^\circ$ from both sides:

$\angle AOD = 180^\circ - 55^\circ$

$\angle AOD = 125^\circ$

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Finding $\angle BOC$:

Similar to finding $\angle AOD$, on line AB, the angles $\angle AOC$ and $\angle BOC$ form a linear pair.

Angles forming a linear pair are supplementary, meaning their sum is $180^\circ$.

Substitute the given value of $\angle AOC$:

Subtract $55^\circ$ from both sides:

$\angle BOC = 180^\circ - 55^\circ$

$\angle BOC = 125^\circ$

---

Alternatively, $\angle AOD$ and $\angle BOC$ are also vertically opposite angles, so they must be equal. Since we found $\angle AOD = 125^\circ$, it follows that $\angle BOC = 125^\circ$, which matches the result obtained using the linear pair property.

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Final Measures:

$\angle BOD = 55^\circ$

$\angle AOD = 125^\circ$

$\angle BOC = 125^\circ$

When a line intersects two or more distinct lines at different points, it is called a transversal. When a transversal intersects two lines, eight angles are formed around the two points of intersection. These angles can be grouped into specific pairs based on their position.

Let's assume a diagram where a transversal line $t$ intersects two lines $l_1$ and $l_2$. The angles formed at the upper intersection are typically labelled $\angle 1, \angle 2, \angle 3, \angle 4$ (e.g., $\angle 1$ top-left, $\angle 2$ top-right, $\angle 3$ bottom-left, $\angle 4$ bottom-right), and the angles formed at the lower intersection are labelled $\angle 5, \angle 6, \angle 7, \angle 8$ in the corresponding positions ($\angle 5$ top-left, $\angle 6$ top-right, $\angle 7$ bottom-left, $\angle 8$ bottom-right).

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(a) Corresponding angles:

Corresponding angles are pairs of angles that are in the same relative position at each intersection where a transversal line intersects two or more lines. They are located on the same side of the transversal and in corresponding positions (either above the lines or below the lines).

Illustration (referring to the standard labelled diagram):

Pairs of corresponding angles are:

- $\angle 1$ and $\angle 5$ (both are in the top-left position at their respective intersections)

- $\angle 2$ and $\angle 6$ (both are in the top-right position)

- $\angle 3$ and $\angle 7$ (both are in the bottom-left position)

- $\angle 4$ and $\angle 8$ (both are in the bottom-right position)

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(b) Alternate interior angles:

Alternate interior angles are pairs of angles that are on opposite sides of the transversal and lie between the two lines that the transversal intersects. The word "interior" refers to the region between the two intersected lines, and "alternate" refers to being on opposite sides of the transversal.

Illustration (referring to the standard labelled diagram, interior angles are $\angle 3, \angle 4, \angle 5, \angle 6$):

Pairs of alternate interior angles are:

- $\angle 3$ and $\angle 6$ (both are interior, on opposite sides of the transversal)

- $\angle 4$ and $\angle 5$ (both are interior, on opposite sides of the transversal)

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(c) Interior angles on the same side of the transversal:

Interior angles on the same side of the transversal (also called consecutive interior angles or same-side interior angles) are pairs of angles that are on the same side of the transversal and lie between the two lines that the transversal intersects.

Illustration (referring to the standard labelled diagram, interior angles are $\angle 3, \angle 4, \angle 5, \angle 6$):

Pairs of interior angles on the same side of the transversal are:

- $\angle 4$ and $\angle 6$ (both are interior, on the same side of the transversal)

- $\angle 3$ and $\angle 5$ (both are interior, on the same side of the transversal)

When a transversal intersects two parallel lines, the angle pairs formed have specific properties.

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1. Corresponding Angles

These are pairs of angles in the same relative position at each intersection where the transversal crosses the parallel lines (e.g., top-left at the first intersection and top-left at the second intersection). If we label the angles, a pair of corresponding angles might be $\angle A$ and $\angle B$.

Property: Corresponding angles are equal.

This means $\angle A = \angle B$ for any pair of corresponding angles.

Explanation: Imagine taking the top parallel line and sliding it downwards along the transversal until it perfectly overlaps the bottom parallel line. Because the lines are parallel, this sliding motion doesn't change the angles. Therefore, the angle at the top intersection would land exactly on top of its corresponding angle at the bottom intersection, showing they must be the same size.

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2. Alternate Interior Angles

These are pairs of angles on opposite sides of the transversal and located between the two parallel lines. If we label the angles, a pair of alternate interior angles might be $\angle C$ and $\angle D$.

Property: Alternate interior angles are equal.

This means $\angle C = \angle D$ for any pair of alternate interior angles.

Explanation: Consider an alternate interior angle, say $\angle C$. Its corresponding angle (say $\angle E$) is equal to it when the lines are parallel ($\angle C = \angle E$). The other alternate interior angle, $\angle D$, is vertically opposite to $\angle E$. Since vertical angles are equal ($\angle D = \angle E$), and $\angle C = \angle E$, it follows that $\angle C = \angle D$.

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3. Alternate Exterior Angles

These are pairs of angles on opposite sides of the transversal and located outside the two parallel lines. If we label the angles, a pair of alternate exterior angles might be $\angle F$ and $\angle G$.

Property: Alternate exterior angles are equal.

This means $\angle F = \angle G$ for any pair of alternate exterior angles.

Explanation: An alternate exterior angle, say $\angle F$, is vertically opposite to a corresponding angle (say $\angle H$). Since vertical angles are equal ($\angle F = \angle H$) and corresponding angles are equal ($\angle G = \angle H$) when the lines are parallel, it follows that $\angle F = \angle G$.

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4. Consecutive Interior Angles (or Same-Side Interior Angles)

These are pairs of angles on the same side of the transversal and located between the two parallel lines. If we label the angles, a pair of consecutive interior angles might be $\angle I$ and $\angle J$.

Property: Consecutive interior angles are supplementary (their measures add up to $180^\circ$).

This means $\angle I + \angle J = 180^\circ$ for any pair of consecutive interior angles.

Explanation: Consider a consecutive interior angle, say $\angle I$. The angle adjacent to it on the straight line formed by the transversal (which is also a corresponding angle to $\angle J$) is supplementary to $\angle I$. Since corresponding angles are equal ($\angle K = \angle J$), substituting $\angle J$ for $\angle K$ in the supplementary relationship ($\angle I + \angle K = 180^\circ$) shows that $\angle I + \angle J = 180^\circ$.

Given that line $m \parallel$ line $n$ and $\angle 1 = 65^\circ$. We need to find the measures of $\angle 2, \angle 3, \angle 4, \angle 5, \angle 6, \angle 7,$ and $\angle 8$.

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Step 1: Find $\angle 2$

$\angle 1$ and $\angle 2$ form a linear pair.

Substituting the given value of $\angle 1$:

So,

---

Step 2: Find $\angle 3$

$\angle 1$ and $\angle 3$ are vertically opposite angles.

So,

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Step 3: Find $\angle 4$

$\angle 2$ and $\angle 4$ are vertically opposite angles.

Using the value of $\angle 2$ found in Step 1:

Alternatively, $\angle 3$ and $\angle 4$ form a linear pair, so $\angle 3 + \angle 4 = 180^\circ$, which gives $65^\circ + \angle 4 = 180^\circ$, hence $\angle 4 = 115^\circ$.

---

Step 4: Find $\angle 5$

$\angle 1$ and $\angle 5$ are corresponding angles.

So,

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Step 5: Find $\angle 6$

$\angle 2$ and $\angle 6$ are corresponding angles.

Using the value of $\angle 2$ found in Step 1:

---

Step 6: Find $\angle 7$

$\angle 3$ and $\angle 7$ are corresponding angles.

Using the value of $\angle 3$ found in Step 2:

---

Step 7: Find $\angle 8$

$\angle 4$ and $\angle 8$ are corresponding angles.

Using the value of $\angle 4$ found in Step 3:

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Alternatively, using other angle properties:

Given $\angle 1 = 65^\circ$ and $m \parallel n$.

• $\angle 4 = 115^\circ$ (Linear pair with $\angle 1$)
• $\angle 3 = 65^\circ$ (Vertically opposite to $\angle 1$)
• $\angle 2 = 115^\circ$ (Vertically opposite to $\angle 4$ or Linear pair with $\angle 1$)

Now for the angles on line $n$:

• $\angle 5 = \angle 1 = 65^\circ$ (Corresponding angles)
• $\angle 6 = \angle 4 = 115^\circ$ (Corresponding angles)
• $\angle 7 = \angle 3 = 65^\circ$ (Corresponding angles)
• $\angle 8 = \angle 2 = 115^\circ$ (Corresponding angles)

Or using alternate angles:

• $\angle 3$ and $\angle 5$ are alternate interior angles, so $\angle 5 = \angle 3 = 65^\circ$.
• $\angle 4$ and $\angle 6$ are alternate interior angles, so $\angle 6 = \angle 4 = 115^\circ$.
• $\angle 1$ and $\angle 7$ are alternate exterior angles, so $\angle 7 = \angle 1 = 65^\circ$.
• $\angle 2$ and $\angle 8$ are alternate exterior angles, so $\angle 8 = \angle 2 = 115^\circ$.

Or using consecutive interior angles:

• $\angle 4$ and $\angle 5$ are consecutive interior angles, so $\angle 4 + \angle 5 = 180^\circ$. $115^\circ + \angle 5 = 180^\circ \implies \angle 5 = 65^\circ$.
• $\angle 3$ and $\angle 6$ are consecutive interior angles, so $\angle 3 + \angle 6 = 180^\circ$. $65^\circ + \angle 6 = 180^\circ \implies \angle 6 = 115^\circ$.

Summary of angle measures:

• $\angle 1 = 65^\circ$ (Given)
• $\angle 2 = 115^\circ$
• $\angle 3 = 65^\circ$
• $\angle 4 = 115^\circ$
• $\angle 5 = 65^\circ$
• $\angle 6 = 115^\circ$
• $\angle 7 = 65^\circ$
• $\angle 8 = 115^\circ$

We want to show that when two lines intersect, the vertically opposite angles formed are equal in measure.

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Let's consider two straight lines, say line AB and line CD, intersecting at a point O. This intersection forms four angles: $\angle AOC$, $\angle COB$, $\angle BOD$, and $\angle DOA$.

The pairs of vertically opposite angles are:

• $\angle AOC$ and $\angle BOD$
• $\angle COB$ and $\angle DOA$

We will prove that $\angle AOC = \angle BOD$. The proof for the other pair is similar.

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Understanding Linear Pairs

A linear pair is a pair of adjacent angles formed when two lines intersect. These angles share a common vertex and a common side, and their non-common sides are opposite rays forming a straight line. The sum of the measures of angles in a linear pair is always $180^\circ$.

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Proof:

Consider the straight line AB and the ray OC standing on it. The angles $\angle AOC$ and $\angle COB$ form a linear pair.

Now consider the straight line CD and the ray OB standing on it. The angles $\angle COB$ and $\angle BOD$ form a linear pair.

We now have two equations where $\angle COB$ is added to another angle to equal $180^\circ$.

From the first equation: $\angle AOC = 180^\circ - \angle COB$

From the second equation: $\angle BOD = 180^\circ - \angle COB$

Since both $\angle AOC$ and $\angle BOD$ are equal to the same quantity ($180^\circ - \angle COB$), they must be equal to each other.

Thus, vertically opposite angles are equal.

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Similarly, by considering the straight line AB and ray OD, we get $\angle AOD + \angle BOD = 180^\circ$. Also, considering the straight line CD and ray OA, we get $\angle AOC + \angle AOD = 180^\circ$. Using the same logic, we can show that $\angle COB = \angle DOA$.

Given:

An angle is $20^\circ$ less than its complement.

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To Find:

The measure of the angle and the measure of its complement.

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Solution:

Let the measure of the angle be $x$ degrees.

Let the measure of its complement be $y$ degrees.

By the definition of complementary angles, their sum is $90^\circ$.

According to the given condition, the angle $x$ is $20^\circ$ less than its complement $y$.

Now we substitute the expression for $x$ from equation (ii) into equation (i):

$(y - 20^\circ) + y = 90^\circ$

$2y - 20^\circ = 90^\circ$

Add $20^\circ$ to both sides:

$2y = 90^\circ + 20^\circ$

$2y = 110^\circ$

Divide both sides by 2:

$y = \frac{110^\circ}{2}$

$y = 55^\circ$

Now substitute the value of $y = 55^\circ$ back into equation (ii) to find $x$:

$x = 55^\circ - 20^\circ$

$x = 35^\circ$

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Answer:

The measure of the angle is $35^\circ$.

The measure of its complement is $55^\circ$.

Given:

An angle is $30^\circ$ more than twice its supplement.

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To Find:

The measure of the angle.

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Solution:

Let the measure of the angle be $x$ degrees.

Let the measure of its supplement be $y$ degrees.

By the definition of supplementary angles, their sum is $180^\circ$.

According to the given condition, the angle $x$ is $30^\circ$ more than twice its supplement $y$.

Now we have a system of two linear equations.

Substitute the expression for $x$ from equation (ii) into equation (i):

$(2y + 30^\circ) + y = 180^\circ$

Combine the terms with $y$:

$3y + 30^\circ = 180^\circ$

Subtract $30^\circ$ from both sides:

$3y = 180^\circ - 30^\circ$

$3y = 150^\circ$

Divide both sides by 3 to find the value of $y$:

$y = \frac{150^\circ}{3}$

$y = 50^\circ$

Now substitute the value of $y = 50^\circ$ back into equation (ii) to find the value of $x$:

$x = 2(50^\circ) + 30^\circ$

$x = 100^\circ + 30^\circ$

$x = 130^\circ$

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Answer:

The measure of the angle is $130^\circ$.

The supplement of the angle is $50^\circ$. (Check: $130^\circ + 50^\circ = 180^\circ$. Also, $130^\circ = 2(50^\circ) + 30^\circ \implies 130^\circ = 100^\circ + 30^\circ \implies 130^\circ = 130^\circ$. The answer is consistent with the given conditions).

Given the figure showing lines $l$ and $m$ intersected by a transversal.

Two consecutive interior angles are shown with measures $110^\circ$ and $70^\circ$.

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We need to determine if lines $l$ and $m$ are parallel.

We know that if two lines are parallel and intersected by a transversal, then the sum of consecutive interior angles is $180^\circ$ (they are supplementary).

Conversely, if the sum of consecutive interior angles formed by a transversal intersecting two lines is $180^\circ$, then the lines are parallel.

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Let's find the sum of the given consecutive interior angles:

Sum $= 110^\circ + 70^\circ$

Sum $= 180^\circ$

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Since the sum of the consecutive interior angles is $180^\circ$, the lines $l$ and $m$ are parallel.

Reason: When a transversal intersects two lines such that the consecutive interior angles are supplementary, the lines are parallel (Converse of Consecutive Interior Angles Theorem).

Given:

Three lines intersect at a point.

Two adjacent angles formed by these lines measure $40^\circ$ and $30^\circ$.

An angle vertically opposite to the sum of these two angles is marked as $x$.

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To Find:

The value of $x$.

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Solution:

In the given figure, the angles measuring $40^\circ$ and $30^\circ$ are shown as adjacent angles around the point of intersection of the three lines.

The sum of these two adjacent angles represents a single angle formed by two of the three intersecting lines.

The sum of these adjacent angles is calculated as:

$40^\circ + 30^\circ = 70^\circ$

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The angle marked as $x$ in the figure is vertically opposite to the angle formed by the sum of the $40^\circ$ and $30^\circ$ angles.

We utilize the property of angles formed by intersecting lines, which states that vertically opposite angles are equal.

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According to this property, the angle $x$ is equal in measure to the angle that is vertically opposite to it.

Since the angle vertically opposite to $x$ is the angle formed by the sum of the $40^\circ$ and $30^\circ$ angles, we have:

$x = (\text{Sum of adjacent angles})$

$x = 70^\circ$

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Therefore, the value of $x$ is $70^\circ$.

Given:

Line AB is parallel to line CD ($AB \parallel CD$).

Transversal PR intersects AB at point P and CD at point R.

$\angle BPQ = 50^\circ$ (where Q is a point on the transversal such that ray PQ forms the angle with ray PB).

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To Find:

The measures of $\angle APQ$ and $\angle PRD$.

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Solution:

Step 1: Find the measure of $\angle APQ$.

Angles $\angle APQ$ and $\angle BPQ$ form a $\underline{\textbf{linear pair}}$ on the straight line AB at point P.

The sum of angles in a linear pair is $180^\circ$.

$\angle APQ + \angle BPQ = 180^\circ$

Substitute the given value of $\angle BPQ$:

$\angle APQ + 50^\circ = 180^\circ$

Subtract $50^\circ$ from both sides:

$\angle APQ = 180^\circ - 50^\circ$

$\angle APQ = 130^\circ$

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Step 2: Find the measure of $\angle PRD$.

Since lines AB and CD are parallel ($AB \parallel CD$) and PR is the transversal, the angles $\angle BPQ$ and $\angle PRD$ are $\underline{\textbf{corresponding angles}}$.

$\underline{\textbf{When two parallel lines are intersected by a transversal, the corresponding angles are equal}}$.

$\angle PRD = \angle BPQ$

Substitute the given value of $\angle BPQ$:

$\angle PRD = 50^\circ$

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Therefore, the measure of $\angle APQ$ is $130^\circ$ and the measure of $\angle PRD$ is $50^\circ$.

Given:

Parallel lines $p$ and $q$ intersected by transversal $r$.

One interior angle on the left side is $105^\circ$.

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To Find:

Measures of all other angles.

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Solution:

Let the transversal intersect line $p$ at point A and line $q$ at point B.

Let the angles formed at A be $\angle 1, \angle 2, \angle 3, \angle 4$ and at B be $\angle 5, \angle 6, \angle 7, \angle 8$, starting from the top-left and going clockwise at each intersection point.

The interior angles on the left side are $\angle 3$ and $\angle 5$. Assume $\angle 3 = 105^\circ$.

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Angles at point A (on line $p$):

Given $\angle 3 = 105^\circ$ (Interior Left).

$\angle 1$ and $\angle 3$ form a linear pair.

$\angle 1 = 180^\circ - \angle 3 = 180^\circ - 105^\circ = 75^\circ$ (Exterior Left)

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$\angle 2$ and $\angle 3$ are vertically opposite angles.

$\angle 2 = \angle 3 = 105^\circ$ (Exterior Right)

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$\angle 4$ and $\angle 1$ are vertically opposite angles.

$\angle 4 = \angle 1 = 75^\circ$ (Interior Right)

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Angles at point B (on line $q$):

Since $p \parallel q$, we use properties of parallel lines and a transversal.

$\angle 5$ and $\angle 1$ are corresponding angles.

$\angle 5 = 75^\circ$ (Interior Left)

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$\angle 6$ and $\angle 2$ are corresponding angles.

$\angle 6 = 105^\circ$ (Interior Right)

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$\angle 7$ and $\angle 3$ are corresponding angles.

$\angle 7 = 105^\circ$ (Exterior Left)

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$\angle 8$ and $\angle 4$ are corresponding angles.

$\angle 8 = 75^\circ$ (Exterior Right)

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Measures of all other angles:

Angles on the left side (other than the given $105^\circ$ interior angle):

Exterior Left ($\angle 1$) = $75^\circ$

Interior Left ($\angle 5$) = $75^\circ$

Exterior Left ($\angle 7$) = $105^\circ$

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Angles on the right side:

Exterior Right ($\angle 2$) = $105^\circ$

Interior Right ($\angle 4$) = $75^\circ$

Interior Right ($\angle 6$) = $105^\circ$

Exterior Right ($\angle 8$) = $75^\circ$