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Chapter 6 Lines And Angles (Concepts)
Welcome to this rigorous re-examination of the fundamental geometric concepts of Lines and Angles. Building upon the intuitive understanding developed in earlier classes, this chapter adopts a more formal approach, laying down precise definitions, axioms, and theorems that form the bedrock for logical deduction and proof in geometry. We will revisit familiar terms but define them with greater clarity and explore the relationships between lines and angles, particularly when lines intersect or run parallel to each other, culminating in foundational proofs regarding triangles.
We begin by solidifying our understanding of the basic building blocks:
- A line segment: A part of a line with two definite endpoints.
- A ray: A part of a line with one endpoint, extending infinitely in one direction.
- A line: Extending infinitely in both directions (often represented with arrows).
- Collinear points: Three or more points lying on the same straight line.
- Non-collinear points: Points that do not all lie on the same straight line.
Here, we introduce foundational assumptions and derive initial theorems. The Linear Pair Axiom states that if a ray stands on a line, then the sum of the two adjacent angles formed is exactly $180^\circ$. Conversely, if the sum of two adjacent angles is $180^\circ$, then their non-common arms must form a straight line. This axiom is fundamental. Using this, we can rigorously prove the theorem stating that if two lines intersect, the vertically opposite angles formed are always equal.
The core of this chapter delves into the properties arising when a transversal (a line that intersects two or more other lines at distinct points) interacts with parallel lines. When a transversal intersects two parallel lines, specific pairs of angles exhibit consistent relationships:
- Corresponding angles are equal.
- Alternate interior angles are equal.
- Alternate exterior angles are equal.
- Consecutive interior angles (or co-interior angles) are supplementary (their sum is $180^\circ$).
Crucially, the converses of these statements are also true and serve as the primary criteria for proving that two lines are parallel. If a transversal intersects two lines such that any *one* of the following conditions is met:
- A pair of corresponding angles is equal, OR
- A pair of alternate interior angles is equal, OR
- A pair of consecutive interior angles is supplementary (sums to $180^\circ$),
These principles involving parallel lines are then applied to prove fundamental properties of triangles. The most significant is the Angle Sum Property of a Triangle, which states that the sum of the measures of the three interior angles of any triangle is always $180^\circ$. The standard proof cleverly involves drawing a line through one vertex parallel to the opposite side and using the properties of alternate interior angles with the transversals formed by the other two sides of the triangle. Building on this, the Exterior Angle Theorem is also revisited and proven: the measure of an exterior angle of a triangle is equal to the sum of the measures of its two non-adjacent interior opposite angles. Throughout this chapter, the emphasis is on using these axioms and proven theorems systematically to deduce unknown angle measures in increasingly complex geometric figures involving intersections, parallel lines, and triangles.
Terms and Definitions Related to Lines and Angles
Building upon the fundamental, undefined terms of point, line, and plane introduced in the previous chapter (Introduction to Euclid's Geometry), we can now define other related geometric concepts more precisely. These definitions are essential for a clear understanding of lines and angles and the theorems associated with them.
Basic Geometric Terms (Defined)
Based on the undefined terms, we define the following:
Line Segment:
A line segment is a part or portion of a line that has two definite endpoints. Unlike a line, which extends infinitely, a line segment has a finite length. The length of the line segment $\overline{AB}$ is the shortest distance between the points A and B. A line segment with endpoints A and B is denoted by $\overline{AB}$ or $AB$.
Ray:
A ray is a part or portion of a line that has only one endpoint and extends infinitely in one direction. The endpoint is called the initial point of the ray. A ray starting at point A and passing through point B (in that order) is denoted by $\overrightarrow{AB}$. Note that $\overrightarrow{AB}$ is different from $\overrightarrow{BA}$.
Collinear Points:
Three or more points are said to be collinear points if they all lie on the same straight line. If points A, B, and C lie on the same line, they are collinear.
Non-collinear Points:
Points that do not lie on the same straight line are called non-collinear points. Any two distinct points are always collinear, as there is a unique line passing through them (Euclid's Postulate 1).
Angle:
An angle is a geometric figure formed when two rays originate from the same common endpoint. The two rays forming the angle are called the arms or sides of the angle, and the common endpoint is called the vertex of the angle. Angles represent the amount of rotation between the two rays around the vertex.
Angles are typically measured in degrees ($^\circ$). A complete rotation is $360^\circ$.
Types of Angles
Angles are classified based on the measure of their rotation. Let $\theta$ represent the measure of an angle.
| Type of Angle | Measure ($\theta$) | Description |
|---|---|---|
| Zero Angle | $\theta = 0^\circ$ | An angle where the two rays forming the angle coincide, with no separation between them. |
| Acute Angle | $0^\circ < \theta < 90^\circ$ | An angle whose measure is greater than $0^\circ$ but less than $90^\circ$. |
| Right Angle | $\theta = 90^\circ$ | An angle whose measure is exactly $90^\circ$. The arms are perpendicular to each other. |
| Obtuse Angle | $90^\circ < \theta < 180^\circ$ | An angle whose measure is greater than $90^\circ$ but less than $180^\circ$. |
| Straight Angle | $\theta = 180^\circ$ | An angle whose measure is exactly $180^\circ$. The arms form a straight line in opposite directions from the vertex. |
| Reflex Angle | $180^\circ < \theta < 360^\circ$ | An angle whose measure is greater than $180^\circ$ but less than $360^\circ$. It is the larger angle formed by two rays from the same vertex. |
| Complete Angle | $\theta = 360^\circ$ | An angle formed when a ray completes a full rotation and returns to its original position. The arms coincide. |
Example 1. Classify the following angles based on their measures: $35^\circ$, $100^\circ$, $90^\circ$, $180^\circ$, $210^\circ$, $0^\circ$, $360^\circ$, $89^\circ$, $181^\circ$.
Answer:
- $35^\circ$: Since $0^\circ < 35^\circ < 90^\circ$, it is an Acute Angle.
- $100^\circ$: Since $90^\circ < 100^\circ < 180^\circ$, it is an Obtuse Angle.
- $90^\circ$: It is exactly $90^\circ$, so it is a Right Angle.
- $180^\circ$: It is exactly $180^\circ$, so it is a Straight Angle.
- $210^\circ$: Since $180^\circ < 210^\circ < 360^\circ$, it is a Reflex Angle.
- $0^\circ$: It is exactly $0^\circ$, so it is a Zero Angle.
- $360^\circ$: It is exactly $360^\circ$, so it is a Complete Angle.
- $89^\circ$: Since $0^\circ < 89^\circ < 90^\circ$, it is an Acute Angle.
- $181^\circ$: Since $180^\circ < 181^\circ < 360^\circ$, it is a Reflex Angle.
Pairs of Angles
In geometry, angles rarely exist in isolation. They are often related to each other based on their position or the sum of their measures. Understanding these special pairs of angles is a fundamental skill for solving geometric problems.
Complementary Angles
Two angles are called complementary if they add up to exactly $90^\circ$ (a right angle). If two angles are complementary, each angle is known as the "complement" of the other.
Example 1. Find the complement of an angle measuring $40^\circ$.
Answer:
Let the measure of the complement be $x$.
By the definition of complementary angles, the sum of an angle and its complement is $90^\circ$.
$40^\circ + x = 90^\circ$
To find $x$, subtract $40^\circ$ from both sides:
$x = 90^\circ - 40^\circ = 50^\circ$
The complement of a $40^\circ$ angle is $50^\circ$.
Supplementary Angles
Two angles are called supplementary if they add up to exactly $180^\circ$ (a straight angle). Each angle is known as the "supplement" of the other.
Example 2. Two supplementary angles are in the ratio $2:7$. Find the measures of the angles.
Answer:
Let the two angles be $2x$ and $7x$.
Since the angles are supplementary, their sum must be $180^\circ$.
$2x + 7x = 180^\circ$
$9x = 180^\circ$
$x = \frac{180^\circ}{9} = 20^\circ$
Now, we find the measure of each angle:
First angle $= 2x = 2 \times 20^\circ = 40^\circ$.
Second angle $= 7x = 7 \times 20^\circ = 140^\circ$.
The two angles are $40^\circ$ and $140^\circ$. (Check: $40^\circ + 140^\circ = 180^\circ$).
Adjacent Angles
Two angles are adjacent if they are "next to each other." For two angles to be adjacent, they must satisfy all three of the following conditions:
- They have a common vertex (corner point).
- They have a common arm (side).
- Their non-common arms lie on opposite sides of the common arm.
In the figure, $\angle 1$ and $\angle 2$ are adjacent angles.
Linear Pair of Angles
A linear pair is a special pair of adjacent angles whose non-common arms form a straight line. Because they form a straight line, the sum of the angles in a linear pair is always $180^\circ$.
In other words, a linear pair consists of two adjacent angles that are supplementary.
Linear Pair Axiom
This is a fundamental assumption in geometry:
Axiom 6.1: If a ray stands on a line, then the sum of the two adjacent angles so formed is $180^\circ$.
Axiom 6.2 (Converse): If the sum of two adjacent angles is $180^\circ$, then their non-common arms form a straight line.
Vertically Opposite Angles
When two lines intersect, they form an 'X' shape. The angles that are directly opposite each other at the intersection point are called vertically opposite angles. They are always equal in measure.
In the figure, the pair $(\angle 1, \angle 3)$ and the pair $(\angle 2, \angle 4)$ are vertically opposite angles.
Theorem: Vertically opposite angles are equal
This is not an assumption but a statement that can be proven using the Linear Pair Axiom.
Theorem 6.1. If two lines intersect each other, then the vertically opposite angles are equal.
Proof:
Given:
Two lines, AB and CD, intersect at a point O.
To Prove:
(i) $\angle AOC = \angle BOD$
(ii) $\angle AOD = \angle BOC$
Proof:
Part (i): Proving $\angle AOD = \angle BOC$
Consider the ray OA standing on the line CD. The angles $\angle AOC$ and $\angle AOD$ form a linear pair.
$\angle AOC + \angle AOD = 180^\circ$
[Linear Pair Axiom] ... (1)
Now, consider the ray OD standing on the line AB. The angles $\angle AOD$ and $\angle BOD$ form a linear pair.
$\angle AOD + \angle BOD = 180^\circ$
[Linear Pair Axiom] ... (2)
From equations (1) and (2), we see that both expressions are equal to $180^\circ$. Therefore, they must be equal to each other.
$\angle AOC + \angle AOD = \angle AOD + \angle BOD$
[Things equal to the same thing are equal]
Subtracting $\angle AOD$ from both sides of the equation:
$\angle AOC = \angle BOD$
This proves that one pair of vertically opposite angles is equal.
Part (ii): Proving $\angle AOD = \angle BOC$
Similarly, we can consider ray OC on line AB and ray OB on line CD.
$\angle AOC + \angle BOC = 180^\circ$
[Linear Pair] ... (3)
$\angle BOD + \angle BOC = 180^\circ$
[Linear Pair] ... (4)
From (3) and (4):
$\angle AOC + \angle BOC = \angle BOD + \angle BOC$
Subtracting $\angle BOC$ from both sides gives:
$\angle AOC = \angle BOD$
Thus, both pairs of vertically opposite angles are equal. Hence, the theorem is proved.
Angles Made by a Transversal with Two Lines
When we have two or more lines in a plane, and another line intersects them at distinct points, this intersecting line is called a transversal. The intersection creates several angles, and the relationships between these angles are important in geometry, especially when the two lines are parallel.
What is a Transversal?
A transversal is a line that intersects two or more distinct lines at distinct points.
In the figure, line $n$ is a transversal intersecting lines $l$ and $m$ at two different points. If line $n$ intersected $l$ and $m$ at the same point, it would just be two lines intersecting at one point, not a transversal cutting two lines.
Angles Formed by a Transversal
When a transversal intersects two lines, a total of eight angles are formed – four angles at each of the two intersection points. Let's number these angles for reference (typically starting from the top-left at the first intersection and going clockwise, then doing the same at the second intersection):
The region between lines $l$ and $m$ is called the interior, and the regions outside lines $l$ and $m$ are called the exterior.
Types of Angle Pairs Formed by a Transversal
The eight angles formed by a transversal can be grouped into several specific pairs based on their positions:
1. Corresponding Angles: These are pairs of angles that are in the same relative position at each intersection. They are on the same side of the transversal, with one angle being an interior angle and the other being an exterior angle. There are four pairs of corresponding angles.
In the figure above, the corresponding angle pairs are:
- $\angle 1$ and $\angle 5$
- $\angle 2$ and $\angle 6$
- $\angle 3$ and $\angle 7$
- $\angle 4$ and $\angle 8$
2. Alternate Interior Angles: These are pairs of interior angles that are on opposite sides of the transversal. They are located between the two lines and do not share a vertex.
In the figure above, the alternate interior angle pairs are:
- $\angle 3$ and $\angle 6$
- $\angle 4$ and $\angle 5$
3. Alternate Exterior Angles: These are pairs of exterior angles that are on opposite sides of the transversal. They are located outside the two lines and do not share a vertex.
In the figure above, the alternate exterior angle pairs are:
- $\angle 1$ and $\angle 8$
- $\angle 2$ and $\angle 7$
4. Interior Angles on the Same Side of the Transversal: These are pairs of interior angles that lie on the same side of the transversal. They are located between the two lines and share a segment of the transversal. They are also called consecutive interior angles or co-interior angles.
In the figure above, the interior angle pairs on the same side of the transversal are:
- $\angle 3$ and $\angle 5$
- $\angle 4$ and $\angle 6$
5. Exterior Angles on the Same Side of the Transversal: These are pairs of exterior angles that lie on the same side of the transversal. They are located outside the two lines and share a segment of the transversal. They are also called consecutive exterior angles or co-exterior angles.
In the figure above, the exterior angle pairs on the same side of the transversal are:
- $\angle 1$ and $\angle 7$
- $\angle 2$ and $\angle 8$
Understanding these pairs and their positions is fundamental for studying the properties of angles formed when a transversal intersects parallel lines.
Parallel Lines and A Transversal
When a transversal cuts across two lines, it creates eight angles. If the two lines are parallel, these eight angles have special, predictable relationships with each other. Understanding these relationships is fundamental to Euclidean geometry, allowing us to prove that lines are parallel or find unknown angle measures.
The Corresponding Angles Axiom
The relationship between corresponding angles and parallel lines is the foundation from which other relationships are proven. It is accepted as an axiom, meaning it is a self-evident truth that does not require proof.
Axiom 6.3 (Corresponding Angles Axiom)
If a transversal intersects two parallel lines, then each pair of corresponding angles is equal.
In the figure above, if $l \parallel m$, then: $\angle 1 = \angle 5$, $\angle 2 = \angle 6$, $\angle 4 = \angle 8$, and $\angle 3 = \angle 7$.
Axiom 6.4 (Converse of Corresponding Angles Axiom)
This is the reverse statement, which gives us a powerful method to prove that two lines are parallel.
If a transversal intersects two lines such that any pair of corresponding angles is equal, then the two lines are parallel to each other.
For example, if you can show that $\angle 1 = \angle 5$, you can conclude that $l \parallel m$.
Theorems on Parallel Lines
Using the Corresponding Angles Axiom and other basic principles (like the Linear Pair Axiom and the Vertically Opposite Angles Theorem), we can prove other key relationships.
Alternate Interior Angles
Theorem 6.2. If a transversal intersects two parallel lines, then each pair of alternate interior angles is equal.
Proof:
Theorem: If a transversal intersects two parallel lines, then each pair of alternate interior angles is equal.
Given: Two parallel lines, $l$ and $m$, are intersected by a transversal line $n$.
To Prove: $\angle 3 = \angle 5$.
Proof:
We are given that $l \parallel m$. By the Corresponding Angles Axiom, we know that corresponding angles are equal.
$\angle 1 = \angle 5$
[Corresponding Angles Axiom] ... (i)
Also, $\angle 1$ and $\angle 3$ are vertically opposite angles formed by the intersection of lines $l$ and $n$. Therefore, they are equal.
$\angle 1 = \angle 3$
[Vertically Opposite Angles] ... (ii)
From equations (i) and (ii), since both $\angle 5$ and $\angle 3$ are equal to $\angle 1$, they must be equal to each other.
$\angle 3 = \angle 5$
Similarly, we can prove $\angle 4 = \angle 6$. Hence, the theorem is proved.
Theorem 6.3 (Converse). If a transversal intersects two lines such that a pair of alternate interior angles is equal, then the two lines are parallel.
Proof:
Theorem: If a transversal intersects two lines such that a pair of alternate interior angles is equal, then the two lines are parallel.
Given: A transversal $n$ intersects two lines $l$ and $m$ such that $\angle 3 = \angle 5$.
To Prove: $l \parallel m$.
Proof:
We are given that alternate interior angles are equal.
$\angle 3 = \angle 5$
[Given] ... (i)
We also know that $\angle 1$ and $\angle 3$ are vertically opposite angles.
$\angle 1 = \angle 3$
[Vertically Opposite Angles] ... (ii)
From (i) and (ii), it follows that $\angle 1 = \angle 5$.
Since $\angle 1$ and $\angle 5$ are corresponding angles and they are equal, by the Converse of the Corresponding Angles Axiom, the lines $l$ and $m$ must be parallel.
Interior Angles on the Same Side of the Transversal
Theorem 6.4. If a transversal intersects two parallel lines, then each pair of interior angles on the same side of the transversal is supplementary (adds up to $180^\circ$).
Proof:
Theorem: If a transversal intersects two parallel lines, then each pair of interior angles on the same side of the transversal is supplementary.
Given: Two parallel lines, $l$ and $m$, are intersected by a transversal line $n$.
To Prove: $\angle 4 + \angle 5 = 180^\circ$.
Proof:
The angles $\angle 4$ and $\angle 2$ form a linear pair on the line $l$. Therefore, their sum is $180^\circ$.
$\angle 4 + \angle 2 = 180^\circ$
[Linear Pair Axiom] ... (i)
Since we are given that $l \parallel m$, the alternate interior angles $\angle 2$ and $\angle 5$ are equal.
$\angle 2 = \angle 5$
[Alternate Interior Angles Theorem] ... (ii)
Substitute the value of $\angle 2$ from equation (ii) into equation (i):
$\angle 4 + \angle 5 = 180^\circ$
This proves that the interior angles on the same side of the transversal are supplementary.
Theorem 6.5 (Converse). If a transversal intersects two lines such that a pair of interior angles on the same side is supplementary, then the two lines are parallel.
Proof:
Theorem: If a transversal intersects two lines such that a pair of interior angles on the same side is supplementary, then the two lines are parallel.
Given: A transversal $n$ intersects two lines $l$ and $m$ such that $\angle 4 + \angle 5 = 180^\circ$.
To Prove: $l \parallel m$.
Proof:
We are given that the interior angles on the same side are supplementary.
$\angle 4 + \angle 5 = 180^\circ$
[Given] ... (i)
We know that $\angle 1$ and $\angle 4$ form a linear pair on the line $l$.
$\angle 1 + \angle 4 = 180^\circ$
[Linear Pair Axiom] ... (ii)
From (i) and (ii), it follows that $\angle 4 + \angle 5 = \angle 1 + \angle 4$.
Subtracting $\angle 4$ from both sides gives $\angle 5 = \angle 1$.
Since $\angle 1$ and $\angle 5$ are corresponding angles and they are equal, by the Converse of the Corresponding Angles Axiom, the lines $l$ and $m$ must be parallel.
Lines Parallel to the Same Line
Theorem 6.6. Lines which are parallel to the same line are parallel to each other.
Proof:
Theorem: Lines which are parallel to the same line are parallel to each other.
Given: Three lines $l, m,$ and $n$ such that $l \parallel m$ and $n \parallel m$.
To Prove: $l \parallel n$.
Construction: Draw a transversal line $p$ that intersects all three lines, creating corresponding angles $\angle 1, \angle 2,$ and $\angle 3$.
Proof:
Since we are given that $l \parallel m$, the corresponding angles are equal.
$\angle 1 = \angle 2$
[Corresponding Angles Axiom] ... (i)
We are also given that $n \parallel m$, so their corresponding angles are also equal.
$\angle 3 = \angle 2$
[Corresponding Angles Axiom] ... (ii)
From (i) and (ii), since both $\angle 1$ and $\angle 3$ are equal to $\angle 2$, they must be equal to each other.
$\angle 1 = \angle 3$
Now, consider the lines $l$ and $n$ with transversal $p$. We have shown that their corresponding angles, $\angle 1$ and $\angle 3$, are equal. By the Converse of the Corresponding Angles Axiom, the lines $l$ and $n$ must be parallel.
Application Examples
Example 1. In the figure, if $AB \parallel CD$, $CD \parallel EF$ and $y:z = 3:7$, find $x$.
Answer:
Given: $AB \parallel CD$ and $CD \parallel EF$. The ratio $y:z = 3:7$.
To Find: The value of $x$.
Solution:
First, since $AB \parallel CD$ and $CD \parallel EF$, by Theorem 6.6, we know that all three lines are parallel to each other: $AB \parallel EF$.
Now consider the parallel lines $AB$ and $EF$. The angles $x$ and $z$ are interior angles on the same side of the transversal. Therefore, their sum is $180^\circ$.
$x + z = 180^\circ$
[Interior angles are supplementary] ... (i)
Next, consider the parallel lines $AB$ and $CD$. The angles $x$ and $y$ are also interior angles on the same side of the transversal.
$x + y = 180^\circ$
[Interior angles are supplementary] ... (ii)
From equations (i) and (ii), we can conclude that $y=z$.
We are given the ratio $y:z = 3:7$. Since $y=z$ seems incorrect from the figure (y is acute, z is obtuse), let's re-examine the angle relationships. Ah, $x$ and $y$ are consecutive interior angles, but $x$ and $z$ are alternate interior angles. Let's correct the approach.
Correct Solution:
Given $AB \parallel CD$, then $x + y = 180^\circ$ (Interior angles). ... (i)
Let the angle vertically opposite to $y$ be $w$. Then $w=y$. Now consider lines $CD \parallel EF$. The angles $w$ and $z$ are interior angles on the same side of the transversal. So, $w+z = 180^\circ$. Since $w=y$, we have:
$y + z = 180^\circ$
[Interior angles are supplementary] ... (ii)
We are given $y:z = 3:7$. Let $y=3k$ and $z=7k$.
Substitute these into equation (ii):
$3k + 7k = 180^\circ \implies 10k = 180^\circ \implies k=18^\circ$.
So, $y = 3k = 3 \times 18^\circ = 54^\circ$.
And $z = 7k = 7 \times 18^\circ = 126^\circ$.
Now, use equation (i) to find $x$:
$x + y = 180^\circ \implies x + 54^\circ = 180^\circ$.
$x = 180^\circ - 54^\circ = 126^\circ$.
The value of $x$ is $126^\circ$.
Example 2. In the figure, if $PQ \parallel ST$, $\angle PQR = 110^\circ$ and $\angle RST = 130^\circ$, find $\angle QRS$.
Answer:
Given: $PQ \parallel ST$, $\angle PQR = 110^\circ$, $\angle RST = 130^\circ$.
To Find: $\angle QRS$.
Construction: Draw a line $XY$ through point $R$ such that $XY \parallel PQ$.
Solution:
Since we constructed $XY \parallel PQ$ and we are given $PQ \parallel ST$, by Theorem 6.6, we have $XY \parallel ST$ as well.
Now, consider the parallel lines $PQ$ and $XY$ and the transversal $QR$. The angles $\angle PQR$ and $\angle QRX$ are interior angles on the same side.
$\angle PQR + \angle QRX = 180^\circ$
$110^\circ + \angle QRX = 180^\circ$
$\angle QRX = 180^\circ - 110^\circ = 70^\circ$.
Next, consider the parallel lines $ST$ and $XY$ and the transversal $SR$. The angles $\angle RST$ and $\angle SRY$ are interior angles on the same side.
$\angle RST + \angle SRY = 180^\circ$
$130^\circ + \angle SRY = 180^\circ$
$\angle SRY = 180^\circ - 130^\circ = 50^\circ$.
From the figure, it is clear that $\angle QRS$ is the sum of the two angles we just found.
$\angle QRS = \angle QRX + \angle SRY$
$\angle QRS = 70^\circ + 50^\circ = 120^\circ$.
The measure of $\angle QRS$ is $120^\circ$.
Angle Properties of a Triangle
A triangle is the simplest possible polygon, defined by three line segments joining three non-collinear points. Despite their simplicity, triangles possess fundamental properties related to their angles that are cornerstones of Euclidean geometry.
Angle Sum Property of a Triangle
This is one of the most well-known and important theorems in geometry. It establishes a fixed relationship among the three interior angles of any triangle.
Theorem 6.7. The sum of the angles of a triangle is $180^\circ$.
Proof:
Given: A triangle PQR with interior angles $\angle 1, \angle 2,$ and $\angle 3$.
To Prove: $\angle 1 + \angle 2 + \angle 3 = 180^\circ$.
Construction: Draw a line XY passing through the vertex P, such that XY is parallel to the base QR ($XY \parallel QR$).
Proof:
Since XY is a straight line, the sum of the angles on the line at point P is $180^\circ$.
$\angle 4 + \angle 1 + \angle 5 = 180^\circ$
[Angles on a straight line] ... (i)
Now, consider the parallel lines XY and QR. PQ is a transversal cutting across them.
The angles $\angle 4$ and $\angle 2$ are a pair of alternate interior angles. Since $XY \parallel QR$, these angles must be equal.
$\angle 4 = \angle 2$
[Alternate Interior Angles] ... (ii)
Similarly, PR is another transversal cutting across the same parallel lines.
The angles $\angle 5$ and $\angle 3$ are also a pair of alternate interior angles. Therefore, they must also be equal.
$\angle 5 = \angle 3$
[Alternate Interior Angles] ... (iii)
Now, substitute the results from (ii) and (iii) into equation (i). Replace $\angle 4$ with $\angle 2$ and $\angle 5$ with $\angle 3$.
$\angle 2 + \angle 1 + \angle 3 = 180^\circ$
Rearranging the terms, we get:
$\angle 1 + \angle 2 + \angle 3 = 180^\circ$
This proves that the sum of the angles of a triangle is always $180^\circ$.
Exterior Angle of a Triangle
When any side of a triangle is extended, the angle formed on the outside is called an exterior angle. The two angles inside the triangle that are not adjacent to this exterior angle are called the interior opposite angles (or remote interior angles).
Theorem 6.8 (Exterior Angle Theorem). If a side of a triangle is produced, then the exterior angle so formed is equal to the sum of the two interior opposite angles.
Proof:
Given: A triangle PQR where the side QR is extended to a point S, forming the exterior angle $\angle PRS$.
To Prove: $\angle PRS = \angle PQR + \angle QPR$.
Proof:
From the Angle Sum Property of a Triangle, we know that the sum of the interior angles of $\triangle PQR$ is $180^\circ$.
$\angle PQR + \angle QPR + \angle PRQ = 180^\circ$
[Angle Sum Property] ... (i)
Also, the angle $\angle PRQ$ and the exterior angle $\angle PRS$ form a linear pair on the straight line QS. Therefore, their sum is also $180^\circ$.
$\angle PRQ + \angle PRS = 180^\circ$
[Linear Pair Axiom] ... (ii)
Since the right sides of equations (i) and (ii) are both $180^\circ$, their left sides must be equal.
$\angle PQR + \angle QPR + \angle PRQ = \angle PRQ + \angle PRS$
Subtracting the common angle $\angle PRQ$ from both sides gives us:
$\angle PQR + \angle QPR = \angle PRS$
This proves that the exterior angle is equal to the sum of the two interior opposite angles.
Important Consequence: Since the angles of a triangle are always positive, this theorem also implies that an exterior angle of a triangle is always greater than either of its interior opposite angles.
Application Examples
Example 1. The angles of a triangle are in the ratio $2:3:4$. Find the measure of each angle.
Answer:
Let the angles of the triangle be $2x, 3x,$ and $4x$.
By the Angle Sum Property, the sum of these angles must be $180^\circ$.
$2x + 3x + 4x = 180^\circ$
$9x = 180^\circ$
$x = \frac{180^\circ}{9} = 20^\circ$
Now we find each angle:
- First angle $= 2x = 2 \times 20^\circ = 40^\circ$
- Second angle $= 3x = 3 \times 20^\circ = 60^\circ$
- Third angle $= 4x = 4 \times 20^\circ = 80^\circ$
The angles of the triangle are $40^\circ, 60^\circ,$ and $80^\circ$.
Example 2. In the given figure, if $QT \perp PR$, $\angle TQR = 40^\circ$ and $\angle SPR = 30^\circ$, find $x$ and $y$.
Answer:
Let's find the values of $x$ and $y$ step-by-step.
To find x:
We focus on the right-angled triangle $\triangle TQR$.
We are given that $QT \perp PR$, which means the angle formed at T is a right angle: $\angle QTR = 90^\circ$.
We are also given that $\angle TQR = 40^\circ$.
The angle $\angle TRQ$ is labeled as $x$.
Using the Angle Sum Property for $\triangle TQR$ (the sum of angles in a triangle is $180^\circ$):
$\angle TQR + \angle QTR + \angle TRQ = 180^\circ$
Substitute the known values:
$40^\circ + 90^\circ + x = 180^\circ$
$130^\circ + x = 180^\circ$
Solve for $x$:
$x = 180^\circ - 130^\circ = 50^\circ$
So, $x = 50^\circ$.
To find y:
The angle $y$ is the exterior angle to $\triangle PQR$ at vertex R.
We can find $y$ using the Exterior Angle Theorem, which states that the exterior angle is equal to the sum of the two interior opposite angles ($\angle QPR$ and $\angle PQR$).
First, we need to find the full measure of $\angle PQR$. We know $\angle PQR = \angle PQT + \angle TQR$. We already know $\angle TQR=40^\circ$, so we need to find $\angle PQT$.
Let's consider the right-angled triangle $\triangle PQT$.
Since $\angle QTR = 90^\circ$ and PR is a straight line, $\angle QTP$ is also $90^\circ$ (Linear Pair).
We are given $\angle QPR = \angle SPR = 30^\circ$.
Using the Angle Sum Property for $\triangle PQT$:
$\angle QPT + \angle QTP + \angle PQT = 180^\circ$
$30^\circ + 90^\circ + \angle PQT = 180^\circ$
$120^\circ + \angle PQT = 180^\circ$
$\angle PQT = 180^\circ - 120^\circ = 60^\circ$
Now we can find the full angle $\angle PQR$:
$\angle PQR = \angle PQT + \angle TQR = 60^\circ + 40^\circ = 100^\circ$
Finally, we apply the Exterior Angle Theorem to find $y$:
$y = \angle QPR + \angle PQR$
[Exterior Angle Theorem]
$y = 30^\circ + 100^\circ = 130^\circ$
So, $y = 130^\circ$.
Alternate Method for y:
We can also find $y$ using the concept of a linear pair. The exterior angle $y$ and the interior angle $x$ form a linear pair on the extended line.
$y + x = 180^\circ$
[Linear Pair Axiom]
Since we already found $x=50^\circ$:
$y + 50^\circ = 180^\circ$
$y = 180^\circ - 50^\circ = 130^\circ$
Both methods give the same result.