NCERT Solutions for Class 8 Maths: Chapter 3 - Understanding Quadrilaterals

Check NCERT Solutions for Class 8 Maths, Chapter 3 - Understanding Quadrilaterals. With this article, you can access solutions of all the questions of this chapter. All the questions of Chapter 3 - Understanding Quadrilaterals (Class 8 Maths NCERT) are important for the preparation of upcoming CBSE Class 8 Maths exam.

NCERT Solutions for Class 8 Maths Chapter 3 - Understanding Quadrilaterals:

EXERCISE 3.2

1. Given here are some figures.

Classify each of them on the basis of the following.

(a) Simple curve

(b) Simple closed curve

(c) Polygon

(d) Convex polygon

(e) Concave polygon

Solutions:

(a) 1, 2, 5, 6, 7

(b) 1, 2, 5, 6, 7

(c) 1, 2

(d) 2

(e) 1

2. How many diagonals does each of the following have?

(a) A convex quadrilateral

(b) A regular hexagon

(c) A triangle

Solutions:

(a) 2 diagonals

(b) 9 diagonals

(c) Zero, a triangle does not have any diagonal in it.

3. What is the sum of the measures of the angles of a convex quadrilateral? Will this property hold if the quadrilateral is not convex? (Make a non-convex quadrilateral and try!)

Solutions:

A convex quadrilateral is made of two triangles (as shown above) and the sum of measures of a convex quadrilateral is 360 degree.

A quadrilateral can be divided into two triangles so the same property also holds for a quadrilateral which is not convex (as shown above). 

4. Examine the table. (Each figure is divided into triangles and the sum of the angles deduced from that.)

What can you say about the angle sum of a convex polygon with number of sides?

(a) 7

(b) 8

(c) 10

(d) n

Solutions:

From the data given in the table, one can infer that the angle sum of a convex polygon of n sides is (n −2) × 180°. So here we have

(a) (7 − 2) × 180° = 900°

(b) (8 − 2) × 180° = 1080°

(c) (10 − 2) × 180° = 1440°

(d) (n − 2) × 180°

5. What is a regular polygon?

State the name of a regular polygon of

(i) 3 sides

(ii) 4 sides

(iii) 6 sides

Solutions:

A polygon with equal sides and equal angles is called a regular polygon.

(i) Equilateral Triangle

(ii) Square

(iii) Regular Hexagon

6. Find the angle measure x in the following figures.

Solutions:

(a)

Sum of the measures of all the interior angles of a quadrilateral is 360°. So,

⇒50° + 130° + 120° + x = 360°

⇒300° + x = 360°

⇒x = 60°

(b)

According to the figure,

90° + a = 180° (Linear pair)

⇒a = 180° − 90° = 90°

As sum of the measures of all interior angles of a quadrilateral is 360 degree so, 

60° + 70° + x + 90° = 360°

⇒220° + x = 360°

⇒x = 140°

(c)

According to the figure,

70 + a = 180° (Linear pair)

⇒a = 110°

60° + b = 180° (Linear pair)

⇒b = 120°

Now sum of the measures of all interior angles of a pentagon is 540°, So,

120° + 110° + 30° + x + x = 540°

⇒260° + 2x = 540°

⇒2x = 280°

⇒ x = 140°

(d) 

We know that sum of the measures of all interior angles of a pentagon is 540°.

5x = 540°

⇒ x = 108°

7.

(a) Find x + y + z

(b) Find x + y + z + w

Solutions:

(a)

x + 90° = 180° (Linear pair)

⇒ x = 90°

z + 30° = 180° (Linear pair)

⇒z = 150°

y = 90° + 30° (Exterior angle theorem)

⇒ y = 120°

⇒ x + y + z = 90° + 120° + 150° = 360°

(b)

We know that the sum of the measures of all interior angles of a quadrilateral is 360 degree

So, a + 60° + 80° + 120° = 360°

⇒a + 260° = 360°

⇒a = 100°

x + 120° = 180° (Linear pair)

⇒x = 60°

y + 80° = 180° (Linear pair)

⇒y = 100°

z + 60° = 180° (Linear pair)

⇒z = 120°

w + 100° = 180° (Linear pair)

⇒w = 80°

As sum of the measures of all interior angles = x + y + z + w = 60° + 100° + 120° + 80° = 360°.

EXERCISE 3.2

1. Find x in the following figures.

Solutions:

The sum of all exterior angles of any polygon is 360º. So,

(a)

125° + 125° + x = 360°

⇒250° + x = 360°

⇒ x = 110°

(b)

60° + 90° + 70° + x + 90° = 360°

⇒310° + x = 360°

⇒x = 50°

2. Find the measure of each exterior angle of a regular polygon of

(i) 9 sides

(ii) 15 sides

Solutions:

(i) As the sum of all exterior angles of the given polygon = 360°

Also, each exterior angle of a regular polygon has the same measure.

So, measure of each exterior angle of a regular polygon of 9 sides = 360°/9 = 40°

(ii) As the sum of all exterior angles of the given polygon = 360°

Also, each exterior angle of a regular polygon has the same measure.

So, measure of each exterior angle of a regular polygon of 15 sides = 360°/15 = 24°

3. How many sides does a regular polygon have if the measure of an exterior angle is 24°?

Solutions:

As, the sum of all exterior angles of the given polygon = 360°

Also, measure of each exterior angle = 24°

Hence, the number of sides of the regular polygon = 360°/24° =15°

4. How many sides does a regular polygon have if each of its interior angles is 165°?

Solutions:

According to the question,

Measure of each interior angle = 165°

So, the measure of each exterior angle = 180° − 165° = 15°

Now, the sum of all exterior angles of any polygon is 360°.

So the number of sides of the polygon = 360°/15° = 24°

5. (a) Is it possible to have a regular polygon with measure of each exterior angle as 22°?

(b) Can it be an interior angle of a regular polygon? Why?

Solutions:

We know that the sum of all exterior angles of all polygons is 360°. Now, in a regular polygon, each exterior angle is of the same measure. 

Also if 360° is a perfect multiple of the given exterior angle, then the given polygon will be possible.

(a) Given, exterior angle = 22°

Now, 360° is not a perfect multiple of 22°. So, such polygon is not possible.

(b) Given, interior angle = 22°

As 360° is not a perfect multiple of 158° so such a polygon is not possible.

 6. (a) What is the minimum interior angle possible for a regular polygon? Why?

(b) What is the maximum exterior angle possible for a regular polygon?

Solutions:

A regular polygon have minimum 3 sides. Also, the exterior angle of this triangle will be the maximum exterior angle possible for any regular polygon.

Now, Exterior angle of an equilateral triangle = 360°/3 = 120°

(a) Minimum interior angle = 180° − 120° = 60°

(b) Maximum possible measure of exterior angle for any polygon is 120°

EXERCISE 3.3

1. Given a parallelogram ABCD. Complete each statement along with the definition or property used.

(i) AD =......

(ii) ÐDCB =......

(iii) OC =......

(iv) m ÐDAB + m ÐCDA =......

Solutions:

(i) Opposite sides are equal in length, So, AD = BC (It’s a parallelogram)

(ii) Opposite angles are equal in measure, So, ∠DCB = ∠DAB (It’s a parallelogram)

(iii) Diagonals bisect each other, so, OC = OA (It’s a parallelogram)

(iv) Adjacent angles are supplementary to each other, so, m∠DAB + m∠CDA =180°(It’s a parallelogram)

2. Consider the following parallelograms. Find the values of the unknowns x, y, z.

Solutions:

(i) x + 100° = 180° (As adjacent angles are supplementary)

⇒ x = 80°

⇒z = x = 80° (As, opposite angles are equal)

⇒y = 100° (As, opposite angles are equal)

(ii) 50° + y = 180° (As, adjacent angles are supplementary)

⇒y = 130°

⇒x = y = 130° (As, opposite angles are equal)

⇒z = x = 130° (As, corresponding angles)

(iii) x = 90° (As, vertically opposite angles)

⇒x + y + 30° = 180° (As, angle sum property of triangles)

⇒120° + y = 180°

⇒y = 60°

⇒z = y = 60° (As, alternate interior angles)

(iv) z = 80° (As, corresponding angles)

⇒y = 80° (As, opposite angles are equal)

⇒x+ y = 180° (As, adjacent angles are supplementary)

⇒x = 180° − 80° = 100°

(v) y = 112° (Opposite angles are equal)

⇒x+ y + 40° = 180° (As angle sum property of triangles)

⇒x + 112° + 40° = 180°

⇒x + 152° = 180°

⇒x = 28°

⇒z = x = 28° (As alternate interior angles)

3. Can a quadrilateral ABCD be a parallelogram if

(i) ÐD + ÐB = 180°?

(ii) AB = DC = 8 cm, AD = 4 cm and BC = 4.4 cm?

(iii) ÐA = 70° and ÐC = 65°?

Solutions:

(i) As, ∠D + ∠B = 180°, so it may or may not be a parllelogram as ther condition are also necssary liek um of the measures of adjacent angles should be 180° and opposite angles should also equal

(ii) No. Its’  because opposite sides BC and AD are of different lengths.

(iii) No. Its’  because opposite angles  C and A have different measures.

4. Draw a rough figure of a quadrilateral that is not a parallelogram but has exactly two opposite angles of equal measure.

Solutions:

This figure is similar to a kite and here angle B and D are equal but its not a parallelogram.

5. The measures of two adjacent angles of a parallelogram are in the ratio 3 : 2. Find the measure of each of the angles of the parallelogram.

Solutions:

Suppose the two adjacent angles, ∠A and ∠B, of parallelogram ABCD are in the ratio of 3:2. 

Let’s say ∠A = 3x and ∠B = 2x

AS the sum of the measures of adjacent angles is 180° for a parallelogram.

⇒∠A + ∠B = 180°

⇒3x + 2x = 180°

⇒5x = 180°

⇒x = 180°/5 = 36°

⇒∠A = ∠C = 3x = 108° (Opposite angles)

⇒∠B = ∠D = 2x = 72° (Opposite angles)

Therefore, the measures of the angles of the parallelogram are 108°, 72°, 108°, and 72°.

6. Two adjacent angles of a parallelogram have equal measure. Find the measure of each of the angles of the parallelogram.

Solutions:

As the sum of adjacent angles = 180°

⇒∠A + ∠B = 180°

⇒2∠A = 180° (∠A = ∠B)

⇒∠A = 90°

⇒∠B = ∠A = 90°

⇒∠C = ∠A = 90° (Opposite angles)

⇒∠D = ∠B = 90° (Opposite angles)

Therefore each angle of the parallelogram measures 90°.

7. The adjacent figure HOPE is a parallelogram. Find the angle measures x, y and z. State the properties you use to find them.

Solutions:

Here, y = 40° (Alternate interior angles)

⇒70° = z + 40° (Corresponding angles)

⇒70° − 40° = z

⇒z = 30°

Now, x + (z + 40°) = 180° (Adjacent pair of angles)

x + 70° = 180°

x = 110°

8. The following figures GUNS and RUNS are parallelograms. Find x and y. (Lengths are in cm)

Solutions:

(i) We know that the lengths of opposite sides of a parallelogram are equal to each other.

⇒GU = SN

⇒3y − 1 = 26

⇒3y = 27

⇒y = 9

⇒SG = NU

⇒3x = 18

x = 6

Hence, the measures of x and y are 6 cm and 9 cm respectively.

(ii) As, diagonals of a parallelogram bisect each other.

⇒y + 7 = 20

⇒y = 13

⇒x + y = 16

⇒x + 13 = 16

⇒x = 3

Therefore, the measures of y and x are 13 cm & 3 cm respectively.

9.

In the above figure both RISK and CLUE are parallelograms. Find the value of x.

Solutions:

Given, adjacent angles of a parallelogram are supplementary.

In the given parallelogram RISK, ∠RKS + ∠ISK = 180°

⇒120° + ∠ISK = 180°

⇒∠ISK = 60°

As, opposite angles of a parallelogram are equal.

In the given parallelogram CLUE, ∠ULC = ∠CEU = 70°

Now, the sum of the measures of all the interior angles of a triangle is 180°.

⇒x + 60° + 70° = 180°

⇒ x = 50°

10. Explain how this figure is a trapezium. Which of its two sides are parallel? (Fig 3.32)

Solutions:

If a transversal line is intersecting two given lines such that the sum of the measures of the angles on the same side of transversal is 180°, then the given two lines will be parallel to each other.

Here, ∠NML + ∠MLK = 180°

Hence, NM||LK

As quadrilateral KLMN has a pair of parallel lines, therefore, it is a trapezium. 

Solutions:

Given. AB || DC

⇒∠B + ∠C = 180° (Angles on the same side of transversal)

⇒120º + ∠C = 180°

⇒∠C = 60°

Solutions:

Here, ∠P + ∠Q = 180° (Angles on the same side of transversal)

⇒∠P + 130° = 180°

⇒∠P = 50°

⇒∠R + ∠S = 180° (Angles on the same side of transversal)

⇒90° + ∠R = 180°

⇒∠S = 90°

Yes. One can also find the measure of m∠P by another method. Here, m∠R and m∠Q are given and after finding m∠S, with the help of the angle sum property of a quadrilateral, one can find m∠P.

EXERCISE 3.4

1. State whether True or False.

(a) All rectangles are squares

(b) All rhombuses are parallelograms

(c) All squares are rhombuses and also rectangles

(d) All squares are not parallelograms.

(e) All kites are rhombuses.

(f) All rhombuses are kites.

(g) All parallelograms are trapeziums.

(h) All squares are trapeziums.

Solutions:

(a) False. All rectangles are not squares but all squares are rectangles.

(b) True. (Opposite sides of a rhombus are equal &  parallel to each other).

(c) True. (As, all squares are also rectangles as each internal angle measures 90° and All squares are rhombuses as all sides of a square are of equal lengths). 

(d) False. (As, all squares are parallelograms as opposite sides are equal & parallel).

(e) False. (As, a kite does not have all sides of the same length).

(f) True.(As, a rhombus also has 2 distinct consecutive pairs of sides of equal length).

(g) True. (As, all parallelograms have a pair of parallel sides).

(h) True. (As, all squares have a pair of parallel sides).

2. Identify all the quadrilaterals that have.

(a) four sides of equal length

(b) four right angles

Solutions:

(a) Rhombus and Square are the quadrilaterals that have 4 sides of equal length.

(b) Square and rectangle are the quadrilaterals that have 4 right angles.

3. Explain how a square is.

(i) a quadrilateral

(ii) a parallelogram

(iii) a rhombus

(iv) a rectangle

Solutions:

(i) It has four sides.

(ii) It has 4 sides and its opposite sides are parallel to each other.

(iii) Its four sides are of the same length.

(iv) Each interior angle measures 90°.

4. Name the quadrilaterals whose diagonals.

(i) bisect each other

(ii) are perpendicular bisectors of each other

(iii) are equal

Solutions:

(i) parallelogram, rhombus, square, and rectangle

(ii) Rhombus and square

(iii) Rectangle and square

5. Explain why a rectangle is a convex quadrilateral.

Solutions:

Two diagonals in a rectangle are always lying in the interior of the rectangle. So, it is a convex quadrilateral.

6. ABC is a right-angled triangle and O is the midpoint of the side opposite to the right angle. Explain why O is equidistant from A, B and C. (The dotted lines are drawn additionally to help you).

Solutions:

If we draw lines AD and DC such that AD||BC, AB||DC the AD = BC, AB = DC

Now, ABCD is a rectangle as opposite sides are parallel and  equal to each other. Also, and all the interior angles are of 90°.

In the given rectangle, diagonals are of equal length and also these bisect each other.

Therefore, AO = OC = BO = OD

Hence, O is equidistant from A, B, and C.