Chapter 3 Understanding Quadrilaterals (Additional Questions)

The correct answer is (C) A simple closed curve made up of line segments.

A polygon is a closed plane figure bounded by three or more line segments. It must be a simple curve (does not cross itself) and must be closed.

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Let's examine the given options:

(A) A circle: A circle is a closed curve, but it is not made up of line segments. Therefore, it is not a polygon.

(B) A curve that crosses itself: This describes a non-simple curve. Polygons must be simple closed curves. Therefore, it is not a polygon.

(C) A simple closed curve made up of line segments: This is the precise definition of a polygon. It is a simple curve, it is closed, and it is composed solely of line segments.

(D) An open curve made up of line segments: This is made up of line segments but is an open curve, not a closed one. Polygons must be closed figures. Therefore, it is not a polygon.

The correct answer is (B) $9$.

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Solution:

A hexagon is a polygon with $6$ sides. Let $n$ be the number of sides of a polygon.

The number of diagonals in a polygon with $n$ sides is given by the formula:

$D = \frac{n(n-3)}{2}$

For a hexagon, $n=6$. Substitute $n=6$ into the formula:

$D = \frac{6(6-3)}{2}$

$D = \frac{6(3)}{2}$

$D = \frac{18}{2}$

$D = 9$

Thus, a hexagon has $9$ diagonals.

The correct answer is (B) $1260^\circ$.

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Solution:

A nonagon is a polygon with $9$ sides.

The number of sides of the polygon is $n=9$.

The sum of the interior angles of a polygon with $n$ sides is given by the formula:

Sum of interior angles $= (n-2) \times 180^\circ$

Substitute the number of sides, $n=9$, into the formula:

Sum $= (9-2) \times 180^\circ$

Sum $= 7 \times 180^\circ$

Sum $= 1260^\circ$

The sum of the interior angles of a nonagon is $1260^\circ$.

The correct answer is (B) $12$.

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Solution:

Let $n$ be the number of sides of the regular polygon.

The measure of each exterior angle of a regular polygon is given by the formula:

Exterior Angle $= \frac{360^\circ}{n}$

We are given that the exterior angle is $30^\circ$. So,

$30^\circ = \frac{360^\circ}{n}$

To find the number of sides, $n$, we can rearrange the formula:

$n = \frac{360^\circ}{30^\circ}$

$n = \frac{360}{30}$

$n = 12$

Therefore, the regular polygon has $12$ sides.

The correct answer is (A) $72^\circ$.

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Solution:

Let the two adjacent angles of the parallelogram be $2x$ and $3x$, since they are in the ratio $2:3$.

We know that adjacent angles in a parallelogram are supplementary, which means their sum is $180^\circ$.

So, we can write the equation:

$2x + 3x = 180^\circ$

Combine the terms on the left side:

$5x = 180^\circ$

Now, solve for $x$:

$x = \frac{180^\circ}{5}$

$x = 36^\circ$

The measures of the two adjacent angles are:

Smaller angle $= 2x = 2 \times 36^\circ = 72^\circ$

Larger angle $= 3x = 3 \times 36^\circ = 108^\circ$

The smaller angle is $72^\circ$.

The correct answer is (A) $75^\circ$.

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Solution:

The sum of the interior angles of a quadrilateral is $360^\circ$.

The angles of the quadrilateral are given as $x^\circ, (x+10)^\circ, (x+20)^\circ,$ and $(x+30)^\circ$.

So, we can write the equation:

$x + (x+10) + (x+20) + (x+30) = 360$

Combine the terms with $x$ and the constant terms:

$(x+x+x+x) + (10+20+30) = 360$

$4x + 60 = 360$

Subtract $60$ from both sides of the equation:

$4x = 360 - 60$

$4x = 300$

Divide both sides by $4$ to solve for $x$:

$x = \frac{300}{4}$

$x = 75$

The value of $x$ is $75^\circ$.

The correct answer is (B) Rhombus and Square.

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Explanation:

Let's consider the properties of the diagonals of the given quadrilaterals:

A parallelogram has diagonals that bisect each other. However, they do not necessarily bisect at right angles.

A rectangle has diagonals that bisect each other and are equal in length. They do not necessarily bisect at right angles.

A rhombus has diagonals that bisect each other and are perpendicular (intersect at right angles). They are not necessarily equal in length.

A square is a special type of parallelogram, rectangle, and rhombus. Its diagonals bisect each other, are equal in length, and are perpendicular (intersect at right angles).

A kite has diagonals that are perpendicular. One diagonal bisects the other, but they do not necessarily bisect each other (unless it's a rhombus or square, which is a special kite).

Therefore, the quadrilaterals whose diagonals bisect each other at right angles are the Rhombus and the Square.

The correct answer is (C) Polygon.

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Explanation:

Let's review the definitions of the given options:

(A) Circle: A circle is a closed curved shape where all points on the boundary are equidistant from a central point. It is a simple closed curve but is not made up of line segments.

(B) Arc: An arc is a part of the circumference of a circle or any other curve. It is usually an open curve.

(C) Polygon: A polygon is defined as a simple closed curve that is formed by joining together a finite sequence of straight line segments (sides).

(D) Sphere: A sphere is a perfectly round geometrical object in three-dimensional space that is the surface of a completely round ball. It is a 3D shape, not a 2D curve.

Based on the definitions, a simple closed curve made up of only line segments is called a polygon.

The correct answer is (B) $75^\circ$.

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Explanation:

A key property of parallelograms is that opposite angles are equal in measure.

Given that one angle of the parallelogram measures $75^\circ$, its opposite angle must have the same measure.

The correct answer is (B) $10 \text{ cm}$.

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Solution:

Let the rhombus be ABCD, and let the diagonals be AC and BD. The diagonals of a rhombus bisect each other at right angles.

Let the diagonals intersect at point O.

Length of diagonal AC = $16$ cm.

Length of diagonal BD = $12$ cm.

Since the diagonals bisect each other, we have:

$AO = OC = \frac{1}{2} \times AC = \frac{1}{2} \times 16 = 8$ cm.

$BO = OD = \frac{1}{2} \times BD = \frac{1}{2} \times 12 = 6$ cm.

Since the diagonals bisect each other at right angles, triangle AOB is a right-angled triangle with the right angle at O ($\angle AOB = 90^\circ$).

The sides of the rhombus are the hypotenuses of the four congruent right-angled triangles formed by the diagonals (e.g., triangle AOB, BOC, COD, DOA).

We can use the Pythagorean theorem in triangle AOB to find the length of the side AB (which is a side of the rhombus).

According to the Pythagorean theorem:

$(AB)^2 = (AO)^2 + (BO)^2$

Substitute the values of AO and BO:

$(AB)^2 = (8)^2 + (6)^2$

$(AB)^2 = 64 + 36$

$(AB)^2 = 100$

To find AB, take the square root of both sides:

$AB = \sqrt{100}$

$AB = 10$

The side length of the rhombus is $10 \text{ cm}$.

The correct answer is (B) All rhombuses are parallelograms.

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Explanation:

Let's analyze each statement:

(A) All rectangles are squares. This is false. A rectangle has four right angles, but its sides do not have to be equal. A square is a special type of rectangle where all sides are equal.

(B) All rhombuses are parallelograms. This is true. A rhombus is defined as a quadrilateral with all four sides of equal length. A property of rhombuses is that opposite sides are parallel, which is the definition of a parallelogram. Therefore, every rhombus is a parallelogram.

(C) All trapeziums are kites. This is false. A trapezium (or trapezoid) is a quadrilateral with at least one pair of parallel sides. A kite is a quadrilateral with two distinct pairs of equal-length sides that are adjacent to each other. These properties do not necessarily overlap.

(D) All parallelograms are rectangles. This is false. A parallelogram has opposite sides parallel and equal, but its angles do not have to be right angles. A rectangle is a special type of parallelogram where all angles are right angles.

Based on the definitions and properties of quadrilaterals, only statement (B) is always true.

The correct answer is (B) $90 \text{ m}$.

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Solution:

The garden is rectangular in shape.

Given:

Length ($l$) = $30 \text{ m}$

Width ($w$) = $15 \text{ m}$

The formula for the perimeter of a rectangle is:

Perimeter ($P$) $= 2 \times (\text{length} + \text{width})$

$P = 2 \times (l + w)$

Substitute the given values into the formula:

$P = 2 \times (30 \text{ m} + 15 \text{ m})$

$P = 2 \times (45 \text{ m})$

$P = 90 \text{ m}$

The perimeter of the rectangular garden is $90 \text{ m}$.

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

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Solution:

We know that the sum of the interior angles of a quadrilateral is $360^\circ$.

Let the four angles of the quadrilateral be $\angle A, \angle B, \angle C,$ and $\angle D$.

We are given three angles: $\angle A = 60^\circ$, $\angle B = 80^\circ$, $\angle C = 100^\circ$.

Let the fourth angle, $\angle D$, be $x^\circ$.

The sum of the angles is:

$\angle A + \angle B + \angle C + \angle D = 360^\circ$

Substitute the given values:

$60^\circ + 80^\circ + 100^\circ + x^\circ = 360^\circ$

Combine the known angles:

$(60 + 80 + 100)^\circ + x^\circ = 360^\circ$

$240^\circ + x^\circ = 360^\circ$

Subtract $240^\circ$ from both sides to find $x$:

$x^\circ = 360^\circ - 240^\circ$

$x^\circ = 120^\circ$

The measure of the fourth angle is $120^\circ$.

The correct answer is (B) $360^\circ$.

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Explanation:

A fundamental property of convex polygons is that the sum of their exterior angles, taking one at each vertex, is always constant, regardless of the number of sides.

This sum is always equal to $360^\circ$.

This property holds true for any convex polygon, whether it has 3 sides (triangle), 4 sides (quadrilateral), or any number of sides $n$.

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

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Explanation:

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

Assertion (A): A square is a special type of rhombus and a special type of rectangle. This statement is True.

A square is a quadrilateral with all four sides equal and all four angles equal to $90^\circ$.

A rhombus is a quadrilateral with all four sides equal. Since a square has all four sides equal, it fits the definition of a rhombus, making it a special case of a rhombus.

A rectangle is a quadrilateral with all four angles equal to $90^\circ$. Since a square has all four angles equal to $90^\circ$, it fits the definition of a rectangle, making it a special case of a rectangle.

Reason (R): A square has all properties of a rhombus (all sides equal) and a rectangle (all angles $90^\circ$). This statement is also True.

As explained above, the defining properties of a square include having all sides equal (a property of rhombuses) and all angles equal to $90^\circ$ (a property of rectangles).

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

Assertion (A) claims a square is a special type of both rhombus and rectangle. Reason (R) provides the properties of a square (all sides equal and all angles $90^\circ$) that make it satisfy the definitions of both a rhombus and a rectangle. These properties are precisely why a square is considered a special case of both shapes.

Therefore, Reason (R) correctly explains why Assertion (A) is true.

The correct answer is (B) $115^\circ$.

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Solution:

In a parallelogram, adjacent angles are supplementary. This means that the sum of the measures of consecutive angles is $180^\circ$.

Angles A and B are adjacent angles in the parallelogram ABCD.

Therefore, we have the relationship:

We are given that $\angle A = 65^\circ$. Substitute this value into the equation (i):

$65^\circ + \angle B = 180^\circ$

To find the measure of $\angle B$, subtract $65^\circ$ from both sides of the equation:

$\angle B = 180^\circ - 65^\circ$

$\angle B = 115^\circ$

Thus, the measure of angle B is $115^\circ$.

The correct answer is (C) Diagonals are equal.

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Explanation:

Let's examine each property to see if it is always true for any parallelogram:

(A) Opposite sides are equal. This is a defining property of a parallelogram. So, this is always true.

(B) Opposite angles are equal. This is also a property of a parallelogram. So, this is always true.

(C) Diagonals are equal. This is NOT always true for a parallelogram. Diagonals are equal only in special types of parallelograms, namely rectangles and squares.

(D) Adjacent angles are supplementary. This is a property of a parallelogram. The sum of measures of adjacent angles is $180^\circ$. So, this is always true.

Therefore, the statement that is NOT always true for a parallelogram is that its diagonals are equal.

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

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Explanation:

Let's match each quadrilateral with its property:

(i) Rhombus: A rhombus is a parallelogram with all four sides equal. Its diagonals are perpendicular bisectors of each other.

Property (b) states "Diagonals are perpendicular bisectors of each other". This matches the property of a rhombus.

(ii) Rectangle: A rectangle is a parallelogram with four right angles. Its diagonals are equal in length and bisect each other.

Property (c) states "Diagonals are equal". This matches a property of a rectangle.

(iii) Trapezium: A trapezium (or trapezoid) is a quadrilateral with at least one pair of parallel sides.

Property (a) states "One pair of parallel sides". This is the defining property of a trapezium.

(iv) Kite: A kite is a quadrilateral with two distinct pairs of adjacent sides that are equal in length. Its diagonals are perpendicular, and the longer diagonal bisects the shorter diagonal (so only one diagonal is bisected by the other in general, unless it's a rhombus/square).

Property (d) states "Diagonals are perpendicular, but only one diagonal is bisected". This matches the property of a kite.

Based on these matches, the correct combination is (i)-(b), (ii)-(c), (iii)-(a), (iv)-(d).

The correct answers are (A) A circle and (C) A triangle.

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Explanation:

A simple curve is a curve that does not cross itself.

A closed curve is a curve that starts and ends at the same point.

A simple closed curve is a closed curve that does not cross itself.

Let's evaluate each option:

(A) A circle: A circle starts and ends at the same point (closed) and does not cross itself (simple). Thus, it is a simple closed curve.

(B) A figure 8 shape: This curve starts and ends at the same point (closed), but it crosses itself in the middle. Thus, it is a closed curve but not a simple curve, and therefore not a simple closed curve.

(C) A triangle: A triangle is a closed figure made up of three line segments. It starts and ends at the same point (closed) and does not cross itself (simple). Thus, it is a simple closed curve (and also a polygon).

(D) A curve that starts and ends at the same point but crosses itself: This is the description of a closed curve that is not simple. Thus, it is not a simple closed curve.

Based on the definitions, only the circle and the triangle fit the description of a simple closed curve.

The correct answer is (B) Both A and R are true, but R is not the correct explanation of A.

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Explanation:

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

Assertion (A): A concave quadrilateral has at least one interior angle greater than $180^\circ$. This statement is True. A concave polygon (and thus a concave quadrilateral) is defined by having at least one interior angle greater than $180^\circ$. This large angle is often called a reflex angle.

Reason (R): A convex quadrilateral has all interior angles less than $180^\circ$ and all diagonals lie entirely inside the figure. This statement is also True. This is the definition and a key property of a convex polygon. All interior angles of a convex polygon are less than $180^\circ$, and all diagonals lie within the boundary of the polygon.

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

Assertion (A) defines a property of a concave quadrilateral. Reason (R) describes the properties of a convex quadrilateral. While concave and convex are related (a polygon is either convex or concave), the properties of convex shapes do not directly explain why a concave shape has an angle greater than $180^\circ$. The reason for the concave angle lies in the definition of concavity itself, not in the properties of convexity.

Therefore, both statements are true, but the Reason does not provide the explanation for the Assertion.

The correct answer is (A) $7500 \text{ m}^2$.

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Given:

Shape of the plot: Trapezium

Length of parallel side $a = 100 \text{ m}$

Length of parallel side $b = 150 \text{ m}$

Perpendicular distance between parallel sides (height) $h = 60 \text{ m}$

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To Find:

Area of the trapezium plot.

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Solution:

The formula for the area of a trapezium is given by:

Area $= \frac{1}{2} \times (\text{sum of parallel sides}) \times \text{height}$

Area $= \frac{1}{2} \times (a+b) \times h$

Substitute the given values of $a$, $b$, and $h$ into the formula:

Area $= \frac{1}{2} \times (100 \text{ m} + 150 \text{ m}) \times 60 \text{ m}$

Area $= \frac{1}{2} \times (250 \text{ m}) \times 60 \text{ m}$

Area $= \frac{1}{2} \times 250 \times 60 \text{ m}^2$

Area $= 125 \times 60 \text{ m}^2$

Let's perform the multiplication:

$\begin{array}{cc}& & 1 & 2 & 5 \\ \times & & & 6 & 0 \\ \hline && 0 & 0 & 0 \\ & 7 & 5 & 0 & \times \\ \hline 7 & 5 & 0 & 0 \\ \hline \end{array}$

Area $= 7500 \text{ m}^2$

The area of the trapezium plot is $7500 \text{ m}^2$.

The correct answer is (A) $\textsf{₹} 80000$.

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Given:

Lengths of parallel sides of the trapezium: $AB = 100 \text{ m}$, $CD = 150 \text{ m}$.

Lengths of non-parallel sides: $AD = 65 \text{ m}$, $BC = 85 \text{ m}$.

Cost of fencing per meter = $\textsf{₹} 200/\text{m}$.

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To Find:

Total cost of fencing the plot.

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Solution:

The cost of fencing the plot depends on its perimeter.

The perimeter of a quadrilateral is the sum of the lengths of its four sides.

Perimeter of trapezium ABCD $= AB + BC + CD + AD$

Substitute the given side lengths:

Perimeter $= 100 \text{ m} + 85 \text{ m} + 150 \text{ m} + 65 \text{ m}$

Calculate the sum of the lengths:

Perimeter $= (100 + 85 + 150 + 65) \text{ m}$

Perimeter $= 400 \text{ m}$

The total cost of fencing is the perimeter multiplied by the cost per meter.

Total Cost $= \text{Perimeter} \times \text{Cost per meter}$

Total Cost $= 400 \text{ m} \times \textsf{₹} 200/\text{m}$

Total Cost $= 400 \times 200 \textsf{₹}$

Total Cost $= 80000 \textsf{₹}$

Let's perform the multiplication (as requested, if absolutely needed, showing it for clarity):

$\begin{array}{cccccc} & & & 4 & 0 & 0 \\ \times & & & & 2 & 0 & 0 \\ \hline & & & 0 & 0 & 0 \\ & & 0 & 0 & 0 & \times \\ 8 & 0 & 0 & \times & \times \\ \hline 8 & 0 & 0 & 0 & 0 \\ \hline \end{array}$

The total cost of fencing the plot is $\textsf{₹} 80000$.

The correct answer is (A) Two distinct pairs of equal consecutive sides.

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Explanation:

Let's examine each property in the context of a kite:

(A) Two distinct pairs of equal consecutive sides. This is a common way to define a kite. However, if the two pairs are not "distinct" (meaning all four sides are equal), the quadrilateral is a rhombus. A rhombus is considered a special type of kite. In a rhombus, all four sides are equal, so there aren't *distinct* pairs of different lengths. Thus, this property is not always true for *all* kites (specifically, it's false for a rhombus which is a kite).

(B) The diagonals are perpendicular to each other. This is a fundamental property of all kites, including rhombuses and squares (which are special kites). This is always true.

(C) One of the diagonals bisects the other. In a kite, the diagonal that is the axis of symmetry (connecting the vertices where the unequal sides meet) is the perpendicular bisector of the other diagonal. Therefore, one diagonal is always bisected by the other. This is always true.

(D) One pair of opposite angles is equal. In a kite, the pair of opposite angles located between the sides of unequal length are always equal. This is always true.

Considering that a rhombus is a type of kite, the statement "Two distinct pairs of equal consecutive sides" is not true for a rhombus. Therefore, it is not always true for all kites.

The correct answers are (A) Parallelogram, (B) Rhombus, (C) Rectangle, and (D) Square.

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Explanation:

Let's examine the property of diagonals for each type of quadrilateral:

(A) Parallelogram: By definition, a parallelogram is a quadrilateral with two pairs of parallel sides. A key property of any parallelogram is that its diagonals bisect each other.

(B) Rhombus: A rhombus is a quadrilateral with all four sides equal. A rhombus is a special type of parallelogram. Since all parallelograms have diagonals that bisect each other, rhombuses also have this property.

(C) Rectangle: A rectangle is a quadrilateral with four right angles. A rectangle is a special type of parallelogram. Since all parallelograms have diagonals that bisect each other, rectangles also have this property.

(D) Square: A square is a quadrilateral with all four sides equal and four right angles. A square is a special type of rhombus and also a special type of rectangle. Both rhombuses and rectangles are parallelograms. Since all parallelograms have diagonals that bisect each other, squares also have this property.

All the listed quadrilaterals (Rhombus, Rectangle, and Square) are specific types of Parallelograms. The property that diagonals bisect each other is a characteristic property of all parallelograms and thus holds true for Rhombuses, Rectangles, and Squares as well.

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

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Solution:

The number of diagonals ($D$) in a polygon with $n$ sides is given by the formula:

$D = \frac{n(n-3)}{2}$

Let's calculate the number of diagonals for each polygon listed:

(i) Pentagon (5 sides): Here $n=5$.

$D = \frac{5(5-3)}{2} = \frac{5(2)}{2} = \frac{10}{2} = 5$. Matches (b).

(ii) Triangle (3 sides): Here $n=3$.

$D = \frac{3(3-3)}{2} = \frac{3(0)}{2} = 0$. Matches (a).

(iii) Quadrilateral (4 sides): Here $n=4$.

$D = \frac{4(4-3)}{2} = \frac{4(1)}{2} = \frac{4}{2} = 2$. Matches (c).

(iv) Octagon (8 sides): Here $n=8$.

$D = \frac{8(8-3)}{2} = \frac{8(5)}{2} = \frac{40}{2} = 20$. Matches (d).

Matching the polygon with the calculated number of diagonals, we get:

(i) Pentagon - 5 (b)

(ii) Triangle - 0 (a)

(iii) Quadrilateral - 2 (c)

(iv) Octagon - 20 (d)

This corresponds to option (A).

The correct answer is (B) $6$.

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Explanation:

The question asks for the number of sides of a hexagonal frame.

By definition, a hexagon is a polygon with six sides.

Therefore, a hexagonal frame has $6$ sides.

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

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Solution:

A regular hexagon has $n=6$ sides.

The sum of the interior angles of a polygon with $n$ sides is given by the formula: $(n-2) \times 180^\circ$.

For a hexagon, the sum of interior angles is $(6-2) \times 180^\circ = 4 \times 180^\circ = 720^\circ$.

Since the hexagon is regular, all interior angles are equal.

The measure of each interior angle of a regular polygon with $n$ sides is given by the formula:

Each Interior Angle $= \frac{\text{Sum of Interior Angles}}{n}$

Each Interior Angle $= \frac{(n-2) \times 180^\circ}{n}$

Substitute $n=6$ for a regular hexagon:

Each Interior Angle $= \frac{(6-2) \times 180^\circ}{6}$

Each Interior Angle $= \frac{4 \times 180^\circ}{6}$

Each Interior Angle $= \frac{720^\circ}{6}$

Each Interior Angle $= 120^\circ$

Alternatively, you can find the exterior angle first. The sum of exterior angles is $360^\circ$. For a regular polygon, each exterior angle is $\frac{360^\circ}{n}$.

Each Exterior Angle $= \frac{360^\circ}{6} = 60^\circ$

Since an interior angle and its adjacent exterior angle form a linear pair (sum is $180^\circ$):

Each Interior Angle $= 180^\circ - \text{Each Exterior Angle}$

Each Interior Angle $= 180^\circ - 60^\circ = 120^\circ$

The measure of each interior angle of the regular hexagonal frame is $120^\circ$.

The correct answer is (B) $15$.

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Solution:

Let $n$ be the number of sides of the regular polygon.

The sum of the exterior angles of any convex polygon is $360^\circ$.

In a regular polygon, all exterior angles are equal in measure.

The measure of each exterior angle of a regular polygon with $n$ sides is given by the formula:

Each Exterior Angle $= \frac{360^\circ}{n}$

We are given that the measure of each exterior angle is $24^\circ$. So, we can set up the equation:

$24^\circ = \frac{360^\circ}{n}$

To find the number of sides, $n$, we can rearrange the equation:

$n = \frac{360^\circ}{24^\circ}$

$n = \frac{360}{24}$

Now, we divide $360$ by $24$:

$n = 15$

The regular polygon has $15$ sides.

The correct answer is (A) $36^\circ$.

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Solution:

The angles of the quadrilateral are in the ratio $1:2:3:4$.

Let the angles be $x^\circ, 2x^\circ, 3x^\circ,$ and $4x^\circ$.

The sum of the interior angles of a quadrilateral is $360^\circ$.

So, we can write the equation:

$x + 2x + 3x + 4x = 360$

Combine the terms on the left side:

$(1+2+3+4)x = 360$

$10x = 360$

Now, solve for $x$ by dividing both sides by $10$:

$x = \frac{360}{10}$

$x = 36$

The value of $x$ is $36$.

The four angles are:

First angle $= x^\circ = 36^\circ$

Second angle $= 2x^\circ = 2 \times 36^\circ = 72^\circ$

Third angle $= 3x^\circ = 3 \times 36^\circ = 108^\circ$

Fourth angle $= 4x^\circ = 4 \times 36^\circ = 144^\circ$

The angles are $36^\circ, 72^\circ, 108^\circ,$ and $144^\circ$.

The smallest angle is the angle with the smallest part in the ratio, which is $x^\circ$.

The smallest angle is $36^\circ$.

The correct answer is (C) A triangle.

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Explanation:

A convex polygon is a polygon where all interior angles are less than $180^\circ$ and all diagonals lie entirely inside the polygon.

Let's examine the options:

(A) A star shape: A typical star shape is a concave polygon. It has interior angles greater than $180^\circ$, and some diagonals lie outside the figure.

(B) A quadrilateral with one angle measuring $200^\circ$: An interior angle measuring $200^\circ$ is greater than $180^\circ$. This is the definition of a concave polygon. Therefore, this is not a convex polygon.

(C) A triangle: A triangle is a polygon with 3 sides. The sum of the interior angles is $180^\circ$. Each interior angle of a triangle must be less than $180^\circ$. A triangle has no diagonals. Therefore, a triangle always satisfies the definition of a convex polygon.

(D) A polygon where at least one diagonal lies partly outside the polygon: This describes a concave polygon. In a convex polygon, all diagonals lie entirely inside the figure.

Based on the definition of a convex polygon, only a triangle among the options is always convex.

The correct answer is (B) $36 \text{ cm}$.

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Given:

Length of one adjacent side of the parallelogram = $8 \text{ cm}$.

Length of the other adjacent side of the parallelogram = $10 \text{ cm}$.

---

To Find:

Perimeter of the parallelogram.

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Solution:

In a parallelogram, opposite sides are equal in length.

If the adjacent sides are $8 \text{ cm}$ and $10 \text{ cm}$, then the four sides of the parallelogram are $8 \text{ cm}, 10 \text{ cm}, 8 \text{ cm},$ and $10 \text{ cm}$ in sequence.

The perimeter of a polygon is the sum of the lengths of all its sides.

For a parallelogram with adjacent sides of lengths $a$ and $b$, the perimeter ($P$) can be calculated using the formula:

$P = 2 \times (a + b)$

Substitute the given lengths of the adjacent sides, $a = 8 \text{ cm}$ and $b = 10 \text{ cm}$, into the formula:

$P = 2 \times (8 \text{ cm} + 10 \text{ cm})$

$P = 2 \times (18 \text{ cm})$

$P = 36 \text{ cm}$

The perimeter of the parallelogram is $36 \text{ cm}$.

The correct answer is (B) No.

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Explanation:

The sum of the interior angles of any quadrilateral is always $360^\circ$.

Let the four interior angles of a quadrilateral be $\alpha_1, \alpha_2, \alpha_3,$ and $\alpha_4$.

The sum of these angles is:

$\alpha_1 + \alpha_2 + \alpha_3 + \alpha_4 = 360^\circ$

If all four angles were obtuse, it would mean that each angle is greater than $90^\circ$.

So, $\alpha_1 > 90^\circ$, $\alpha_2 > 90^\circ$, $\alpha_3 > 90^\circ$, and $\alpha_4 > 90^\circ$.

If we add these inequalities, the sum of the angles must be greater than the sum of the lower bounds:

$\alpha_1 + \alpha_2 + \alpha_3 + \alpha_4 > 90^\circ + 90^\circ + 90^\circ + 90^\circ$

$\alpha_1 + \alpha_2 + \alpha_3 + \alpha_4 > 360^\circ$

This result, that the sum of the angles is greater than $360^\circ$, contradicts the known property that the sum of the interior angles of a quadrilateral is exactly $360^\circ$.

Therefore, it is impossible for a quadrilateral to have all four interior angles obtuse.

The correct answer is (A) All sides are equal.

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Explanation:

Let the rhombus be ABCD. In a rhombus, all four sides are equal in length. This is property (A).

Suppose one angle of the rhombus is $60^\circ$. Let $\angle A = 60^\circ$. Since opposite angles in a parallelogram (and thus a rhombus) are equal, $\angle C = 60^\circ$.

The adjacent angles are supplementary, so $\angle B = \angle D = 180^\circ - 60^\circ = 120^\circ$.

The shorter diagonal in this case connects the vertices with the larger angles, which are B and D. So, the shorter diagonal is BD.

Consider the triangle ABD formed by the shorter diagonal BD and two adjacent sides AB and AD.

In a rhombus, AB = AD (property A: All sides are equal).

Therefore, $\triangle ABD$ is an isosceles triangle with vertex angle $\angle A = 60^\circ$.

In any isosceles triangle, the base angles are equal. Let $\angle ABD = \angle ADB$.

The sum of angles in a triangle is $180^\circ$. So, in $\triangle ABD$:

$\angle A + \angle ABD + \angle ADB = 180^\circ$

$60^\circ + \angle ABD + \angle ABD = 180^\circ$ (since $\angle ABD = \angle ADB$)

$60^\circ + 2 \angle ABD = 180^\circ$

$2 \angle ABD = 180^\circ - 60^\circ$

$2 \angle ABD = 120^\circ$

$\angle ABD = 60^\circ$

Since $\angle ABD = \angle ADB = 60^\circ$ and $\angle A = 60^\circ$, all three angles of $\triangle ABD$ are $60^\circ$. This means $\triangle ABD$ is an equilateral triangle.

Similarly, consider $\triangle CBD$. CB = CD (property A). $\angle C = 60^\circ$. $\triangle CBD$ is an isosceles triangle with a $60^\circ$ vertex angle, and thus it is also an equilateral triangle.

The property of the rhombus that directly makes $\triangle ABD$ and $\triangle CBD$ isosceles (AB=AD and CB=CD), which then become equilateral due to the $60^\circ$ angle, is that All sides are equal.

While other properties (like diagonals bisecting angles - C) are also true for a rhombus and contribute to its geometry, the property of having all sides equal is the fundamental reason that, when combined with a $60^\circ$ angle, results in the formation of equilateral triangles by the diagonal connecting the vertices with the $60^\circ$ angles.

The correct answer is (C) The diagonals bisect each other.

---

Explanation:

Let's evaluate each statement about a trapezium:

(A) It has exactly one pair of parallel sides. This is the definition of a standard trapezium (or trapezoid in American English). Some definitions allow "at least one pair", which would include parallelograms, but typically in this context, "exactly one pair" is implied for distinguishing it from a parallelogram. So, this statement is generally considered True for a non-parallelogram trapezium.

(B) The sum of adjacent angles on non-parallel sides is $180^\circ$. This is True. If you have a trapezium with parallel sides AB and CD, then the consecutive interior angles on the transversal AD (i.e., $\angle A$ and $\angle D$) are supplementary ($180^\circ$), and the consecutive interior angles on the transversal BC (i.e., $\angle B$ and $\angle C$) are supplementary ($180^\circ$). These are the angles adjacent to the non-parallel sides.

(C) The diagonals bisect each other. This is FALSE for a general trapezium. Diagonals bisect each other only in parallelograms. A trapezium is not a parallelogram (as it has only one pair of parallel sides, not two). In a trapezium, the diagonals do not bisect each other, except in degenerate cases or if it were a parallelogram (which it isn't by definition in this context).

(D) An isosceles trapezium has equal non-parallel sides and equal base angles. This is True. An isosceles trapezium is a special type of trapezium where the non-parallel sides are equal in length. A property of isosceles trapeziums is that the angles on each base are equal (base angles are equal).

Therefore, the statement that is false about a trapezium is that its diagonals bisect each other.

Definition of a Polygon:

A polygon is a simple closed curve made up of only line segments.

It is a closed plane figure, formed by connecting a sequence of line segments.

---

Difference from a Simple Closed Curve that is Not a Polygon:

A simple closed curve is a continuous curve that starts and ends at the same point and does not cross itself.

A polygon is a *specific type* of simple closed curve.

The key difference lies in the components of the curve:

- A polygon is made up exclusively of straight line segments.

- A simple closed curve that is not a polygon may include curved parts or may be formed by a combination of straight segments and curved parts.

For example, a circle, an ellipse, or a shape made of some straight lines and a curved arc are simple closed curves, but they are not polygons because they contain curved parts.

Convex Polygon:

A convex polygon is a polygon in which all interior angles are less than $180^\circ$.

Alternatively, for any convex polygon, if you draw a line segment connecting any two points inside the polygon, the entire line segment will lie completely inside the polygon.

Another property is that all the diagonals of a convex polygon lie entirely inside the polygon.

Rough Sketch (Example - Convex Quadrilateral):

$\begin{array}{ccc} A & - & B \\ | & & | \\ D & - & C \end{array}$

(This represents a general quadrilateral shape where all corners point outwards)

---

Concave Polygon:

A concave polygon (also known as a non-convex polygon) is a polygon in which at least one interior angle is greater than $180^\circ$.

If you draw a line segment connecting two points inside a concave polygon, it is possible for a part of the line segment to lie outside the polygon.

Also, at least one diagonal of a concave polygon lies partly or entirely outside the polygon.

Concave polygons have at least one vertex that "points inward".

Rough Sketch (Example - Concave Quadrilateral):

$\begin{array}{ccc} A & - & B \\ | & & \\ D & - & C \\ & | & \\ & E & \end{array}$

(This represents a shape like an arrowhead, where angle at C is > 180 degrees, or angle at D is > 180 degrees - imagining D-C-E form the 'caved in' part)

---

Key Difference:

The main difference is that a convex polygon has all interior angles less than $180^\circ$, while a concave polygon has at least one interior angle greater than $180^\circ$. Convex polygons "bulge outwards", while concave polygons have at least one "dent" or "cave in".

Sum of Interior Angles of a Polygon with $n$ Sides:

The sum of the interior angles of a polygon with $n$ sides is given by the formula:

where $n$ is the number of sides of the polygon.

---

Sum of Interior Angles of a Hexagon:

A hexagon is a polygon with $6$ sides.

So, for a hexagon, $n=6$.

Using the formula from (i):

Thus, the sum of the interior angles of a hexagon is $720^\circ$.

The sum of the measures of the exterior angles of any convex polygon is always constant, regardless of the number of sides of the polygon.

The sum of the exterior angles of any convex polygon is $360^\circ$.

Definition of a Quadrilateral:

A quadrilateral is a polygon that has four sides.

It is a simple closed figure formed by four line segments.

---

Number of Sides, Angles, and Vertices:

A quadrilateral has:

- Four sides.

- Four interior angles.

- Four vertices (corners).

For example, in a quadrilateral ABCD, the sides are AB, BC, CD, and DA; the interior angles are $\angle$A, $\angle$B, $\angle$C, and $\angle$D; and the vertices are A, B, C, and D.

Definition of a Parallelogram:

A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel.

If ABCD is a parallelogram, then AB is parallel to DC, and AD is parallel to BC.

---

Properties of a Parallelogram:

Here are two properties of a parallelogram regarding its sides and angles:

1. Regarding sides:

The opposite sides of a parallelogram are equal in length.

If ABCD is a parallelogram, then AB = DC and AD = BC.

2. Regarding angles:

The opposite angles of a parallelogram are equal in measure.

If ABCD is a parallelogram, then $\angle$A = $\angle$C and $\angle$B = $\angle$D.

Given:

The angles of a quadrilateral are in the ratio $1:2:3:4$.

---

To Find:

The measure of each angle of the quadrilateral.

---

Solution:

Let the angles of the quadrilateral be $x^\circ$, $2x^\circ$, $3x^\circ$, and $4x^\circ$, according to the given ratio.

The sum of the interior angles of a quadrilateral is $360^\circ$.

Therefore, we can write the equation:

$x + 2x + 3x + 4x = 360$

Combining the terms on the left side:

$10x = 360$

To find the value of $x$, divide both sides by 10:

$x = \frac{360}{10}$

$x = 36$

Now, we can find the measure of each angle:

First angle = $x^\circ = 36^\circ$

Second angle = $2x^\circ = 2 \times 36^\circ = 72^\circ$

Third angle = $3x^\circ = 3 \times 36^\circ = 108^\circ$

Fourth angle = $4x^\circ = 4 \times 36^\circ = 144^\circ$

The measures of the angles are $36^\circ$, $72^\circ$, $108^\circ$, and $144^\circ$.

Let's verify the sum:

$36^\circ + 72^\circ + 108^\circ + 144^\circ = 108^\circ + 108^\circ + 144^\circ = 216^\circ + 144^\circ = 360^\circ$. The sum is $360^\circ$, which is correct for a quadrilateral.

Given:

ABCD is a parallelogram.

$\angle A = 75^\circ$.

---

To Find:

The measures of $\angle B$ and $\angle D$.

---

Solution:

In a parallelogram, opposite angles are equal.

Therefore, $\angle A = \angle C$ and $\angle B = \angle D$.

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

Also, consecutive angles in a parallelogram are supplementary (their sum is $180^\circ$).

So, $\angle A + \angle B = 180^\circ$ and $\angle A + \angle D = 180^\circ$.

Using $\angle A + \angle B = 180^\circ$:

Subtract $75^\circ$ from both sides:

Since opposite angles are equal, $\angle D = \angle B$.

From (i), we have:

Alternatively, using $\angle A + \angle D = 180^\circ$:

Subtract $75^\circ$ from both sides:

And since $\angle B = \angle D$, $\angle B = 105^\circ$.

Thus, the measures of $\angle B$ and $\angle D$ are $105^\circ$ each.

Definition of a Rhombus:

A rhombus is a parallelogram in which all four sides are equal in length.

In other words, it is a quadrilateral with four equal sides.

---

Properties of a Rhombus Not Necessarily True for a General Parallelogram:

Here are two properties of a rhombus that distinguish it from a general parallelogram:

1. All sides are equal: In a rhombus, all four sides are congruent. A general parallelogram only requires opposite sides to be equal.

2. Diagonals are perpendicular bisectors of each other: The diagonals of a rhombus intersect at a right angle ($90^\circ$) and bisect each other. In a general parallelogram, the diagonals only bisect each other; they are not necessarily perpendicular.

Definition of a Rectangle:

A rectangle is a parallelogram in which all four interior angles are right angles ($90^\circ$).

Alternatively, it is a quadrilateral with four right angles.

---

Properties of a Rectangle Not Necessarily True for a General Parallelogram:

Here are two properties of a rectangle that distinguish it from a general parallelogram:

1. All angles are right angles: In a rectangle, each interior angle measures $90^\circ$. A general parallelogram only requires opposite angles to be equal and consecutive angles to be supplementary; they are not necessarily $90^\circ$.

2. Diagonals are equal in length: The diagonals of a rectangle are congruent. In a general parallelogram, the diagonals only bisect each other; their lengths are not necessarily equal unless it is a rectangle or a square.

Definition of a Square:

A square is a quadrilateral that has all four sides equal in length and all four interior angles equal to $90^\circ$ (right angles).

It can also be defined as a rectangle with all sides equal, or a rhombus with all angles equal to $90^\circ$.

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Why a Square is a Special Type of Rectangle:

A rectangle is defined as a parallelogram with four right angles.

A square meets this definition because:

- It is a parallelogram (since opposite sides are parallel and equal).

- It has four right angles ($90^\circ$).

Therefore, a square is a rectangle. What makes it *special* is the additional property that all its sides are equal, which is not required for a general rectangle (where only opposite sides need to be equal).

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Why a Square is a Special Type of Rhombus:

A rhombus is defined as a parallelogram with four equal sides.

A square meets this definition because:

- It is a parallelogram (since opposite sides are parallel and equal).

- It has four equal sides.

Therefore, a square is a rhombus. What makes it *special* is the additional property that all its interior angles are right angles ($90^\circ$), which is not required for a general rhombus (where only opposite angles need to be equal).

In summary, a square possesses all the properties of both a rectangle and a rhombus, making it a special case of both shapes.

Given:

The lengths of the diagonals of a rectangle are given as $2x+5$ and $3x-4$.

---

To Find:

The value of $x$ and the length of each diagonal.

---

Solution:

A property of a rectangle is that its diagonals are equal in length.

Therefore, we can set the two given expressions for the lengths of the diagonals equal to each other:

$2x+5 = 3x-4$

To solve for $x$, we can rearrange the equation.

Subtract $2x$ from both sides of the equation:

$5 = 3x - 2x - 4$

$5 = x - 4$

Add 4 to both sides of the equation:

$5 + 4 = x$

$9 = x$

The value of $x$ is $9$.

Now, we need to find the length of each diagonal by substituting the value of $x$ into either of the given expressions.

Using the first expression for the length of a diagonal:

Length $= 2x + 5$

Substitute $x=9$:

Length $= 2(9) + 5$

Length $= 18 + 5$

Length $= 23$

Using the second expression for the length of a diagonal:

Length $= 3x - 4$

Substitute $x=9$:

Length $= 3(9) - 4$

Length $= 27 - 4$

Length $= 23$

Both expressions give the same length, confirming the value of $x$ is correct.

So, the value of $x$ is $9$ and the length of each diagonal is $23$.

Definition of a Trapezium:

A trapezium (or trapezoid in some regions) is a quadrilateral with at least one pair of opposite sides parallel.

In a trapezium ABCD, if AB is parallel to DC, then AB and DC are the parallel sides, and AD and BC are the non-parallel sides.

---

Isosceles Trapezium:

A trapezium is called an isosceles trapezium if its non-parallel sides are equal in length.

So, in a trapezium ABCD where AB is parallel to DC, it is an isosceles trapezium if AD = BC.

Additionally, in an isosceles trapezium, the base angles are equal. That is, the angles on the same base are equal (e.g., $\angle$D = $\angle$C if DC is the base, and $\angle$A = $\angle$B if AB is the base).

Definition of a Kite:

A kite is a quadrilateral that has two distinct pairs of adjacent sides of equal length.

In a kite ABCD, where AB is adjacent to BC, and CD is adjacent to DA, this means AB = BC and CD = DA.

---

Properties of a Kite:

Here are two properties of a kite regarding its sides and diagonals:

1. Regarding sides:

There are two distinct pairs of adjacent sides that are equal in length.

(As per the definition, e.g., AB = BC and CD = DA).

2. Regarding diagonals:

The diagonals are perpendicular to each other.

Also, one of the diagonals (the one connecting the vertices where unequal sides meet) is the perpendicular bisector of the other diagonal.

Yes, a rhombus can be a square.

---

A rhombus is a parallelogram with all four sides equal in length.

A square is a quadrilateral with four equal sides and four right angles ($90^\circ$).

A square is also a type of parallelogram, a type of rectangle, and a type of rhombus.

For a rhombus to be a square, it must satisfy the properties of a square that are not necessarily true for a rhombus. The distinguishing property of a square among rhombuses is the measure of its angles.

The condition under which a rhombus is a square is when all its interior angles are equal to $90^\circ$ (right angles).

In a rhombus, opposite angles are equal, and consecutive angles are supplementary (sum up to $180^\circ$). If one angle of a rhombus is $90^\circ$, then its opposite angle is also $90^\circ$. Since consecutive angles are supplementary, the adjacent angles must also be $180^\circ - 90^\circ = 90^\circ$. Thus, if one angle of a rhombus is $90^\circ$, all its angles are $90^\circ$, making it a square.

So, a rhombus is a square if and only if it has at least one right angle.

Yes, a rectangle can be a square.

---

A rectangle is a parallelogram with four right angles ($90^\circ$).

A square is a quadrilateral with four equal sides and four right angles ($90^\circ$).

A square is also a type of parallelogram, a type of rectangle, and a type of rhombus.

For a rectangle to be a square, it must satisfy the properties of a square that are not necessarily true for a rectangle. The distinguishing property of a square among rectangles is the length of its sides.

The condition under which a rectangle is a square is when all its sides are equal in length.

In a rectangle, opposite sides are equal. If a rectangle has all its sides equal, it means that its adjacent sides are also equal (since all sides are equal). A figure with four equal sides and four $90^\circ$ angles is a square.

So, a rectangle is a square if and only if its adjacent sides are equal in length.

Is a Rhombus a Parallelogram?

Yes, a rhombus is a parallelogram.

A parallelogram is defined as a quadrilateral in which both pairs of opposite sides are parallel.

A rhombus is defined as a parallelogram in which all four sides are equal in length.

Since the definition of a rhombus explicitly states that it is a parallelogram, it inherently possesses all the properties of a parallelogram, including having opposite sides parallel and equal.

---

Is a Parallelogram a Rhombus?

No, a general parallelogram is not necessarily a rhombus.

While a parallelogram has opposite sides parallel and equal, it does not require all four sides to be of equal length.

A rhombus, by definition, must have all four sides equal in length.

For example, a rectangle is a parallelogram, but it is only a rhombus if its adjacent sides are equal, making it a square. A non-square rectangle is a parallelogram but not a rhombus.

Therefore, a parallelogram is a rhombus only if it has the additional property that all its sides are equal in length.

Given:

The sum of the interior angles of a regular polygon is $1080^\circ$.

---

To Find:

The number of sides of the polygon.

---

Solution:

Let $n$ be the number of sides of the regular polygon.

The sum of the interior angles of a polygon with $n$ sides is given by the formula:

We are given that the sum of the interior angles is $1080^\circ$.

So, we have the equation:

$(n-2) \times 180^\circ = 1080^\circ$

To solve for $n$, divide both sides by $180^\circ$:

$\frac{(n-2) \times 180^\circ}{180^\circ} = \frac{1080^\circ}{180^\circ}$

$n-2 = \frac{1080}{180}$

$n-2 = 6$

Add 2 to both sides of the equation:

$n = 6 + 2$

$n = 8$

The number of sides of the polygon is $8$.

A polygon with 8 sides is called an octagon.

Given:

ABCD is a parallelogram.

One angle of the parallelogram is $65^\circ$. Let's assume $\angle A = 65^\circ$.

---

To Find:

The measures of the other three angles ($\angle B$, $\angle C$, and $\angle D$).

---

Solution:

In a parallelogram, opposite angles are equal.

Therefore, the angle opposite to $\angle A$ is $\angle C$, and they are equal.

Also, consecutive angles in a parallelogram are supplementary (their sum is $180^\circ$).

So, the angle consecutive to $\angle A$ (which is $\angle B$ or $\angle D$) will sum up to $180^\circ$ with $\angle A$.

Let's find $\angle B$ using the property of consecutive angles:

Substitute the value of $\angle A$:

Subtract $65^\circ$ from both sides:

Finally, the angle opposite to $\angle B$ is $\angle D$, and they are equal.

The measures of the other three angles are $115^\circ$, $65^\circ$, and $115^\circ$.

Given:

Lengths of the diagonals of a rhombus are $6$ cm and $8$ cm.

---

To Find:

The length of each side of the rhombus.

---

Solution:

Let the rhombus be ABCD, and let its diagonals be AC and BD.

Assume the length of diagonal AC is $6$ cm and the length of diagonal BD is $8$ cm.

Let the diagonals intersect at point O.

One of the key properties of a rhombus is that its diagonals bisect each other at right angles ($90^\circ$).

Since the diagonals bisect each other, the segments formed are half the length of the diagonals.

Length of AO = OC = $\frac{1}{2} \times \text{Length of AC}$

Length of AO = $\frac{1}{2} \times 6$ cm $= 3$ cm

Length of BO = OD = $\frac{1}{2} \times \text{Length of BD}$

Length of BO = $\frac{1}{2} \times 8$ cm $= 4$ cm

Since the diagonals intersect at right angles, the triangle formed by two half-diagonals and a side of the rhombus is a right-angled triangle.

Consider the right-angled triangle AOB (where $\angle$AOB = $90^\circ$).

By the Pythagorean theorem, the square of the hypotenuse (the side of the rhombus, AB) is equal to the sum of the squares of the other two sides (AO and BO).

$AB^2 = AO^2 + BO^2$

Substitute the values of AO and BO:

$AB^2 = (3 \text{ cm})^2 + (4 \text{ cm})^2$

$AB^2 = 9 \text{ cm}^2 + 16 \text{ cm}^2$

$AB^2 = 25 \text{ cm}^2$

To find the length of AB, take the square root of both sides:

$AB = \sqrt{25 \text{ cm}^2}$

$AB = 5$ cm

Since all sides of a rhombus are equal, the length of each side of the rhombus is $5$ cm.

Given:

Base of the parallelogram = $10$ cm.

Corresponding altitude = $6$ cm.

---

To Find:

The area of the parallelogram.

---

Solution:

The area of a parallelogram is calculated using the formula:

Area = Base $\times$ Corresponding Altitude

Substitute the given values into the formula:

Area = $10 \text{ cm} \times 6 \text{ cm}$

Area = $60 \text{ cm}^2$

The area of the parallelogram is $60$ square centimeters.

To Find:

The measure of each interior angle of a regular octagon.

---

Solution:

A regular octagon is a polygon with $8$ sides of equal length and $8$ interior angles of equal measure.

The number of sides is $n = 8$.

The sum of the interior angles of a polygon with $n$ sides is given by the formula:

For an octagon ($n=8$), the sum of the interior angles is:

In a regular polygon, all interior angles are equal.

To find the measure of each interior angle, we divide the sum of interior angles by the number of sides (or angles):

Performing the division:

The measure of each interior angle of a regular octagon is $135^\circ$.

Given:

Measure of one exterior angle of a regular polygon $= 40^\circ$.

---

To Find:

The number of sides of the polygon.

---

Solution:

Let $n$ be the number of sides of the regular polygon.

The sum of the measures of the exterior angles of any convex polygon is always $360^\circ$.

In a regular polygon, all exterior angles are equal.

The measure of each exterior angle of a regular polygon with $n$ sides is given by the formula:

We are given that each exterior angle is $40^\circ$.

So, we can set up the equation:

To solve for $n$, we can rearrange the equation:

The polygon has $9$ sides.

(A regular polygon with 9 sides is called a regular nonagon).

Given:

The adjacent angles of a parallelogram are in the ratio $2:3$.

---

To Find:

The measure of each angle of the parallelogram.

---

Solution:

Let the two adjacent angles of the parallelogram be $2x$ degrees and $3x$ degrees, according to the given ratio.

In a parallelogram, adjacent angles are supplementary, meaning their sum is $180^\circ$.

Therefore, we can write the equation:

Combining the terms on the left side:

$5x = 180$

To find the value of $x$, divide both sides by 5:

$x = \frac{180}{5}$

$x = 36$

Now we can find the measure of the two adjacent angles:

First angle = $2x^\circ = 2 \times 36^\circ = 72^\circ$

Second angle = $3x^\circ = 3 \times 36^\circ = 108^\circ$

In a parallelogram, opposite angles are equal.

So, the four angles of the parallelogram are the two adjacent angles and their opposite angles.

The angles are $72^\circ$, $108^\circ$, $72^\circ$, and $108^\circ$.

Let's verify the sum of the interior angles:

$72^\circ + 108^\circ + 72^\circ + 108^\circ = 180^\circ + 180^\circ = 360^\circ$. This is correct for a quadrilateral.

Angle Sum Property of a Convex Quadrilateral:

The angle sum property of a convex quadrilateral states that the sum of the measures of the interior angles of any convex quadrilateral is always $360^\circ$.

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Proof using Angle Sum Property of a Triangle:

Consider a convex quadrilateral ABCD.

Rough Sketch:

$\begin{array}{ccc} A & - & B \\ | \quad \ddots & & | \\ D & - & C \end{array}$

Draw a diagonal, say AC, which divides the quadrilateral into two triangles: $\triangle$ABC and $\triangle$ADC.

Rough Sketch with Diagonal:

$\begin{array}{ccc} A & - & B \\ | \quad \diagdown & & | \\ D & - & C \end{array}$

In $\triangle$ABC, by the angle sum property of a triangle, the sum of its interior angles is $180^\circ$.

In $\triangle$ADC, by the angle sum property of a triangle, the sum of its interior angles is $180^\circ$.

Now, consider the angles of the quadrilateral ABCD:

$\angle$A = $\angle$BAC + $\angle$DAC

$\angle$B = $\angle$ABC

$\angle$C = $\angle$BCA + $\angle$DCA

$\angle$D = $\angle$ADC

The sum of the interior angles of the quadrilateral ABCD is:

$\angle$A + $\angle$B + $\angle$C + $\angle$D = $(\angle$BAC + $\angle$DAC) + $\angle$ABC + $(\angle$BCA + $\angle$DCA) + $\angle$ADC

Rearranging the terms, we can group the angles of the two triangles:

$\angle$A + $\angle$B + $\angle$C + $\angle$D = ($\angle$BAC + $\angle$ABC + $\angle$BCA) + ($\angle$DAC + $\angle$ADC + $\angle$DCA)

From equation (i), the sum of angles in $\triangle$ABC is $180^\circ$.

From equation (ii), the sum of angles in $\triangle$ADC is $180^\circ$.

Substituting these values:

$\angle$A + $\angle$B + $\angle$C + $\angle$D = $180^\circ + 180^\circ$

$\angle$A + $\angle$B + $\angle$C + $\angle$D = $360^\circ$

Thus, the sum of the interior angles of a convex quadrilateral is $360^\circ$.

Part 1: Find the measure of the fourth angle

Given:

Measures of three angles of a quadrilateral are $80^\circ, 95^\circ,$ and $110^\circ$.

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To Find:

The measure of the fourth angle.

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Solution:

Let the three given angles be $\angle 1 = 80^\circ$, $\angle 2 = 95^\circ$, and $\angle 3 = 110^\circ$.

Let the fourth angle be $\angle 4$.

The sum of the interior angles of a quadrilateral is $360^\circ$.

So, we have:

Substitute the given values:

Summing the three angles:

Subtract $285^\circ$ from both sides to find $\angle 4$:

The measure of the fourth angle is $75^\circ$.

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Part 2: Explanation regarding consecutive angles in a parallelogram

Explanation:

In a parallelogram, a key property is that consecutive angles are supplementary. This means the sum of the measures of any two consecutive angles in a parallelogram is $180^\circ$.

We are given three angles: $80^\circ, 95^\circ,$ and $110^\circ$.

If these three angles were consecutive angles in a parallelogram, let's call them $\angle A, \angle B,$ and $\angle C$ in that order. Then they must satisfy the supplementary property for consecutive pairs.

Let $\angle A = 80^\circ$, $\angle B = 95^\circ$, $\angle C = 110^\circ$.

According to the property of parallelograms:

$\angle A + \angle B$ must equal $180^\circ$.

Check: $80^\circ + 95^\circ = 175^\circ$.

This shows that $80^\circ$ and $95^\circ$ cannot be consecutive angles in a parallelogram.

Also, $\angle B + \angle C$ must equal $180^\circ$.

Check: $95^\circ + 110^\circ = 205^\circ$.

This shows that $95^\circ$ and $110^\circ$ cannot be consecutive angles in a parallelogram.

Since assuming these three angles ($80^\circ, 95^\circ, 110^\circ$) are consecutive leads to a contradiction with the property that consecutive angles of a parallelogram are supplementary, these three angles cannot be consecutive angles of a parallelogram.

Properties of Diagonals:

1. Parallelogram:

In a general parallelogram, the diagonals bisect each other.

This means that the point where the two diagonals intersect is the midpoint of each diagonal. However, the diagonals are not necessarily equal in length, and they do not necessarily intersect at right angles.

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2. Rhombus:

A rhombus is a special type of parallelogram (where all sides are equal).

In a rhombus, the diagonals have the following properties:

- They bisect each other (inheriting the property from being a parallelogram).

- They are perpendicular to each other.

- They bisect the angles of the rhombus through which they pass.

The diagonals are not necessarily equal in length (unless the rhombus is also a square).

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3. Rectangle:

A rectangle is a special type of parallelogram (where all angles are right angles).

In a rectangle, the diagonals have the following properties:

- They bisect each other (inheriting the property from being a parallelogram).

- They are equal in length.

The diagonals do not necessarily intersect at right angles (unless the rectangle is also a square).

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How Diagonal Properties Help Distinguish:

The specific properties of the diagonals beyond simple bisection are key to differentiating these quadrilaterals:

- If a quadrilateral's diagonals merely bisect each other, it is a parallelogram.

- If a parallelogram's diagonals bisect each other AND are perpendicular, it is a rhombus.

- If a parallelogram's diagonals bisect each other AND are equal in length, it is a rectangle.

- If a parallelogram's diagonals bisect each other AND are both perpendicular AND equal in length, it is a square (which is both a rhombus and a rectangle).

Thus, by examining whether the diagonals are equal in length or perpendicular, we can determine if a parallelogram is a rectangle, a rhombus, or possibly a square.

Given:

ABCD is a parallelogram with diagonals AC and BD intersecting at O.

Length of AO = $2x - 5$

Length of OC = $x + 4$

Length of BO = $y - 2$

Length of OD = $3y - 10$

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To Find:

The value of $x$.

The value of $y$.

The lengths of diagonals AC and BD.

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Solution:

In a parallelogram, the diagonals bisect each other.

This means that the point of intersection O is the midpoint of both diagonals AC and BD.

Therefore, the segments of each diagonal are equal in length:

Using the property AO = OC, we can set up an equation for $x$:

$2x - 5 = x + 4$

Subtract $x$ from both sides:

$2x - x - 5 = 4$

$x - 5 = 4$

Add 5 to both sides:

$x = 4 + 5$

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Using the property BO = OD, we can set up an equation for $y$:

$y - 2 = 3y - 10$

Subtract $y$ from both sides:

$-2 = 3y - y - 10$

$-2 = 2y - 10$

Add 10 to both sides:

$-2 + 10 = 2y$

$8 = 2y$

Divide both sides by 2:

$y = \frac{8}{2}$

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Now, we can find the length of diagonal AC. Since O is the midpoint of AC, AC = AO + OC = 2 * AO (or 2 * OC).

Let's use the expression for AO and the value $x=9$:

Length of AO = $2x - 5 = 2(9) - 5 = 18 - 5 = 13$

Length of OC = $x + 4 = 9 + 4 = 13$ (verified, AO = OC)

Length of diagonal AC = AO + OC = $13 + 13 = 26$

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Next, we find the length of diagonal BD. Since O is the midpoint of BD, BD = BO + OD = 2 * BO (or 2 * OD).

Let's use the expression for BO and the value $y=4$:

Length of BO = $y - 2 = 4 - 2 = 2$

Length of OD = $3y - 10 = 3(4) - 10 = 12 - 10 = 2$ (verified, BO = OD)

Length of diagonal BD = BO + OD = $2 + 2 = 4$

The value of $x$ is $9$, the value of $y$ is $4$, the length of diagonal AC is $26$, and the length of diagonal BD is $4$.

Given:

Two angle expressions of a parallelogram: $(3x - 2)^\circ$ and $(50 - x)^\circ$.

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To Find:

The measure of each angle of the parallelogram.

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Solution:

In a parallelogram, there are two possibilities for the relationship between the two given angles:

Case 1: The given angles are adjacent angles.

Case 2: The given angles are opposite angles.

Let's examine Case 1.

Case 1: Adjacent Angles

If the angles are adjacent, they are supplementary. This is a property of parallelograms: consecutive angles are supplementary.

Combining like terms:

$3x - x - 2 + 50 = 180$

$2x + 48 = 180$

Subtract 48 from both sides:

$2x = 180 - 48$

$2x = 132$

Divide by 2:

$x = \frac{132}{2}$

$x = 66$

Now substitute $x = 66$ back into the angle expressions:

First angle = $(3x - 2)^\circ = (3(66) - 2)^\circ = (198 - 2)^\circ = 196^\circ$

Second angle = $(50 - x)^\circ = (50 - 66)^\circ = -16^\circ$

An angle in a polygon cannot have a negative measure, and an interior angle of a parallelogram (which is convex) cannot be greater than $180^\circ$. Therefore, the given angles cannot be adjacent.

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Case 2: Opposite Angles

If the angles are opposite, they are equal in measure. This is a property of parallelograms: opposite angles are equal.

Add $x$ to both sides:

$3x + x - 2 = 50$

$4x - 2 = 50$

Add 2 to both sides:

$4x = 50 + 2$

$4x = 52$

Divide by 4:

$x = \frac{52}{4}$

$x = 13$

Now substitute $x = 13$ back into the angle expressions:

First angle = $(3x - 2)^\circ = (3(13) - 2)^\circ = (39 - 2)^\circ = 37^\circ$

Second angle = $(50 - x)^\circ = (50 - 13)^\circ = 37^\circ$

Since these are opposite angles, two angles of the parallelogram measure $37^\circ$.

In a parallelogram, consecutive angles are supplementary (sum to $180^\circ$).

Let the angles be $A, B, C, D$. If $\angle A = \angle C = 37^\circ$.

Then $\angle A + \angle B = 180^\circ$.

Subtract $37^\circ$ from both sides:

$\angle B = 180^\circ - 37^\circ$

$\angle B = 143^\circ$

Since opposite angles are equal, $\angle D = \angle B$.

The four angles of the parallelogram are $37^\circ, 143^\circ, 37^\circ,$ and $143^\circ$.

Finding the Number of Sides from Interior Angle:

To find the number of sides of a regular polygon given the measure of one interior angle, we can use the relationship between the interior angle and the exterior angle at each vertex.

At each vertex of a polygon, the interior angle and its corresponding exterior angle form a linear pair. This means their sum is $180^\circ$.

So, if we know the interior angle, we can find the measure of the exterior angle:

A key property of any convex polygon is that the sum of its exterior angles (one at each vertex) is always $360^\circ$.

In a regular polygon, all exterior angles are equal in measure.

If the regular polygon has $n$ sides, it also has $n$ exterior angles, all equal.

So, the measure of each exterior angle is $\frac{360^\circ}{n}$.

Therefore, if we find the measure of one exterior angle using the given interior angle, we can find the number of sides ($n$) by rearranging the formula:

First, calculate the exterior angle, then divide $360^\circ$ by that value to get the number of sides.

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Applying to the given problem:

Given:

Interior angle of a regular polygon $= 144^\circ$.

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To Find:

The number of sides of the polygon and the measure of each exterior angle.

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Solution:

First, find the measure of each exterior angle.

So, the measure of each exterior angle is $36^\circ$.

Now, use the formula to find the number of sides, $n$:

The number of sides of the regular polygon is $10$.

The measure of each exterior angle is $36^\circ$.

Part 1: Find the Area of the Field

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Given:

Length of parallel side $a = 18$ meters.

Length of parallel side $b = 30$ meters.

Perpendicular distance between parallel sides (height) $h = 12$ meters.

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To Find:

The area of the trapezium-shaped field.

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Solution:

The area of a trapezium is given by the formula:

Substitute the given values into the formula:

Area $= 24 \text{ m} \times 12 \text{ m}$

The area of the field is $288$ square meters.

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Part 2: Additional Information Required for Fencing Cost

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Given:

Cost of fencing per meter = $\textsf{₹}30$.

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To Identify:

Additional information required to find the total cost of fencing.

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Explanation:

Fencing is done along the boundary (perimeter) of the field.

The total cost of fencing is calculated by multiplying the perimeter of the field by the cost per meter.

Total Cost = Perimeter $\times$ Cost per meter

The perimeter of a quadrilateral (like a trapezium) is the sum of the lengths of all four sides.

We are given the lengths of the two parallel sides ($18$ m and $30$ m).

However, we are not given the lengths of the two non-parallel sides of the trapezium.

To calculate the perimeter, we need the lengths of these non-parallel sides.

Therefore, the additional information required to find the total cost of fencing is the lengths of the two non-parallel sides of the trapezium.

Given:

Length of one diagonal of a rhombus ($d_1$) = $12$ cm.

Length of the side of the rhombus ($s$) = $10$ cm.

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To Find:

The length of the other diagonal ($d_2$).

The area of the rhombus.

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Solution:

Let the rhombus be ABCD, and let the diagonals be AC and BD. Let the diagonals intersect at point O.

We are given that one diagonal is $12$ cm. Let AC $= 12$ cm.

The diagonals of a rhombus bisect each other at right angles.

So, O is the midpoint of AC, and O is the midpoint of BD. The angle $\angle$AOB is $90^\circ$.

Length of AO $= \text{OC} = \frac{1}{2} \times \text{AC} = \frac{1}{2} \times 12 \text{ cm} = 6$ cm.

We are given that the side length of the rhombus is $10$ cm. So, AB $= \text{BC} = \text{CD} = \text{DA} = 10$ cm.

Consider the right-angled triangle AOB (since diagonals are perpendicular, $\angle$AOB $= 90^\circ$).

In $\triangle$AOB, AB is the hypotenuse, and AO and BO are the legs.

By the Pythagoras property (Pythagorean theorem), the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Substitute the known values:

Subtract 36 from both sides to find $\text{BO}^2$:

Take the square root of both sides to find the length of BO:

Since O is the midpoint of diagonal BD, the length of the diagonal BD is twice the length of BO.

So, the length of the other diagonal is $16$ cm.

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Now, find the area of the rhombus.

The area of a rhombus can be calculated using the formula:

where $d_1$ and $d_2$ are the lengths of the two diagonals.

We have $d_1 = 12$ cm and $d_2 = 16$ cm.

The area of the rhombus is $96$ square centimeters.

Statement 1: Every Square is a Rectangle, but not Every Rectangle is a Square

Definition of a Rectangle: A rectangle is a parallelogram with four right angles.

Definition of a Square: A square is a quadrilateral with four equal sides and four right angles.

Explanation (Square is a Rectangle):

A square has four right angles, which is a requirement for a rectangle.

Also, a square has opposite sides parallel (since all sides are equal and opposite sides are equal), which means it is a parallelogram.

Since a square is a parallelogram with four right angles, it fits the definition of a rectangle.

Therefore, every square is a rectangle because it possesses all the defining properties of a rectangle.

Explanation (Not every Rectangle is a Square):

A rectangle is a parallelogram with four right angles.

However, the definition of a rectangle only requires opposite sides to be equal. It does not require all four sides to be equal.

A square, on the other hand, requires all four sides to be equal.

Consider a rectangle with unequal adjacent sides (e.g., length 5 units, width 3 units). This is a rectangle because it has four right angles and is a parallelogram. However, it is not a square because its adjacent sides are not equal.

Therefore, not every rectangle is a square; a rectangle is only a square if it has the additional property that all its sides are equal.

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Statement 2: Every Rhombus is a Parallelogram, but not Every Parallelogram is a Rhombus

Definition of a Parallelogram: A parallelogram is a quadrilateral with both pairs of opposite sides parallel.

Definition of a Rhombus: A rhombus is a parallelogram in which all four sides are equal in length.

Explanation (Rhombus is a Parallelogram):

The definition of a rhombus explicitly states that it is "a parallelogram".

Since a rhombus is defined as a type of parallelogram, it automatically satisfies the defining property of a parallelogram, which is having both pairs of opposite sides parallel.

Therefore, every rhombus is a parallelogram.

Explanation (Not every Parallelogram is a Rhombus):

A parallelogram has both pairs of opposite sides parallel and equal in length.

However, a parallelogram does not require all four sides to be equal in length.

A rhombus requires all four sides to be equal in length.

Consider a parallelogram with unequal adjacent sides (e.g., a non-square rectangle or a parallelogram that is not a rhombus). This figure is a parallelogram because its opposite sides are parallel. However, it is not a rhombus because its adjacent sides (and thus all four sides) are not equal in length.

Therefore, not every parallelogram is a rhombus; a parallelogram is only a rhombus if it has the additional property that all its sides are equal in length.

Given:

Lengths of the unequal diagonals of a kite ($d_1, d_2$) are $16$ cm and $10$ cm.

One of the angles where the unequal sides meet is $100^\circ$.

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To Find:

The area of the kite.

What can be said about the angles where the equal sides meet.

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Solution:

Let the lengths of the diagonals be $d_1 = 16$ cm and $d_2 = 10$ cm.

One property of a kite is that its diagonals are perpendicular to each other.

The area of a kite can be calculated using the formula based on its diagonals:

Substitute the given values:

The area of the kite is $80$ square centimeters.

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Angles of the Kite:

Let the kite be ABCD, with AB=AD and CB=CD. The angles where the unequal sides meet are $\angle$B (between AB and CB) and $\angle$D (between AD and CD). The angles where the equal sides meet are $\angle$A (between AB and AD) and $\angle$C (between CB and CD).

A property of a kite is that the angles between the unequal sides are equal.

So, in our kite ABCD, $\angle$B = $\angle$D.

We are given that one of these angles is $100^\circ$. Therefore, both angles are $100^\circ$.

The sum of the interior angles of any quadrilateral is $360^\circ$.

For kite ABCD, the sum of angles is $\angle$A + $\angle$B + $\angle$C + $\angle$D = $360^\circ$.

Substitute the known angle measures:

Subtract $200^\circ$ from both sides:

The angles where the equal sides meet are $\angle$A and $\angle$C.

What can we say about these angles?

- Their sum is $160^\circ$.

- In a general kite (where it is not a rhombus or square), these two angles ($\angle$A and $\angle$C) are generally unequal.

- The diagonal connecting the vertices where equal sides meet (diagonal AC in this case) bisects both $\angle$A and $\angle$C.

So, we can say their sum is $160^\circ$, they are generally unequal, and they are bisected by the main diagonal.

Given:

Length of the wire = $120$ cm.

The wire is bent to form a regular hexagon.

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To Find:

The length of each side of the hexagon.

The measure of each interior angle of the regular hexagon.

The measure of each exterior angle of the regular hexagon.

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Solution:

A regular hexagon is a polygon with 6 sides of equal length and 6 equal interior angles.

The length of the wire used to form the hexagon represents its perimeter.

The perimeter of a regular polygon is the sum of the lengths of its sides.

Let $s$ be the length of each side of the regular hexagon.

Number of sides of a hexagon, $n = 6$.

Perimeter of the hexagon = Number of sides $\times$ Length of each side

Perimeter = $n \times s$

We are given the perimeter (length of the wire) = $120$ cm.

So, we have:

To find $s$, divide both sides by 6:

The length of each side of the regular hexagon is $20$ cm.

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Now, find the measure of each interior angle.

The sum of the interior angles of a polygon with $n$ sides is given by the formula: $(n-2) \times 180^\circ$.

For a hexagon, $n=6$.

Sum of interior angles $= (6-2) \times 180^\circ = 4 \times 180^\circ = 720^\circ$.

In a regular polygon, all interior angles are equal. To find the measure of each interior angle, divide the sum by the number of sides (or angles).

The measure of each interior angle of the regular hexagon is $120^\circ$.

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Finally, find the measure of each exterior angle.

The sum of the exterior angles of any convex polygon is $360^\circ$.

In a regular polygon, all exterior angles are equal.

The measure of each exterior angle is given by the formula: $\frac{360^\circ}{n}$.

For a hexagon, $n=6$.

Alternatively, the interior angle and exterior angle at each vertex are supplementary.

The measure of each exterior angle of the regular hexagon is $60^\circ$.

Part 1: Rectangle

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Given:

Perimeter of the rectangle = $80$ cm.

Ratio of length to breadth ($l:b$) = $3:2$.

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To Find:

The length and breadth of the rectangle.

The area of the rectangle.

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Solution:

Let the length of the rectangle be $3x$ cm and the breadth be $2x$ cm, based on the given ratio $3:2$.

The formula for the perimeter of a rectangle is:

Substitute the given perimeter and the expressions for length and breadth:

Simplify the equation:

$80 = 2 \times (5x)$

$80 = 10x$

Divide both sides by 10 to solve for $x$:

Now, calculate the length and breadth using the value of $x$:

Length $= 3x = 3 \times 8 = 24$ cm.

Breadth $= 2x = 2 \times 8 = 16$ cm.

The length of the rectangle is $24$ cm and the breadth is $16$ cm.

The formula for the area of a rectangle is:

Substitute the calculated length and breadth:

The area of the rectangle is $384$ square centimeters.

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Part 2: Square formed by the same wire

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Given:

The same wire is used, so the perimeter of the square is equal to the perimeter of the rectangle.

Perimeter of the square = Perimeter of the rectangle = $80$ cm.

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To Find:

The side length of the square.

The area of the square.

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Solution:

Let the side length of the square be $s$ cm.

The formula for the perimeter of a square is:

Substitute the given perimeter:

Divide both sides by 4 to solve for $s$:

The side length of the square is $20$ cm.

The formula for the area of a square is:

Substitute the calculated side length:

The area of the square is $400$ square centimeters.

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Part 3: Comparison of Areas

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Area of the rectangle = $384 \text{ cm}^2$.

Area of the square = $400 \text{ cm}^2$.

Comparing the two areas: $400 \text{ cm}^2 > 384 \text{ cm}^2$.

The area of the square is greater than the area of the rectangle formed using the same length of wire (same perimeter).