Contents

NCERT Solutions for Class 8 Maths Chapter 3 Understanding Quadrilaterals are part of NCERT Solutions for Class 8 Maths. Here we have given NCERT Solutions for Class 8 Maths Chapter 3 Understanding Quadrilaterals.

Board |
CBSE |

Textbook |
NCERT |

Class |
Class 8 |

Subject |
Maths |

Chapter |
Chapter 3 |

Chapter Name |
Understanding Quadrilaterals |

Exercise |
Ex 3.1, Ex 3.2, Ex 3.3, Ex 3.4 |

Number of Questions Solved |
31 |

Category |
NCERT Solutions |

## NCERT Solutions for Class 8 Maths Chapter 3 Understanding Quadrilaterals

### Chapter 3 Understanding Quadrilaterals Exercise 3.1

**Question 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

**Solution.**

The classification of the given figures is as under :

**(a)** Simple curve : (1), (2), (5), (6), (7) and (8)

**(b)** Simple closed curve : (1), (2), (5), (6) and (7)

**(c)** Polygon: (1) and (2)

**(d)** Convex polygon : (2)

**(e)** Concave polygon : (1) and (4)

**Question 2.**

How many diagonals does each of the following have ?

**(a)** A convex quadrilateral

**(b)** A regular hexagon

**(c)** A triangle.

**Solution.**

**(a)** A convex quadrilateral has two diagonals.

**(b)** A regular hexagon has nine diagonals.

**(c)** A triangle has no diagonal.

**Question 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 j try!)

**Solution.**

The sum of measures of the angles of a convex quadrilateral is 360°. Yes, this property holds in case of the quadrilateral is not convex.

**Question 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

**Solution.**

From the given table, dearly we observe that the sum of angles (interior angles) of a polygon with n sides = (n – 2) x 180°.

**(a)** n = 7

∴ The sum of the angles of a polygon of 7 sides

Angle sum =

=

**(b)** n = 8

∴ The sum of the angles of a polygon of 8 sides

Angle sum =

=

**(c)** n = 10

∴ The sum of the angles of a polygon of 10 sides

Angle sum =

=

**(d)** Clearly from the given table it is observed that the number of triangles is two less them the number of sides in the polygon.

∴ If the polygon has n sides, the number of triangles formed will be (n – 2).

Since the sum of angles of a triangle = 180°

∴ The sum of angles of a polygon of n sides = (n – 2) x 180°.

**Question 5.**

What is a regular polygon ? State the name of a regular polygon of

**(i)** 3 slides

**(ii)** 4 slides

**(iii)** 6 slides

**Solution.**

**A polygon is said to be a regular polygon, if all its**

- interior angles are equal;
- sides are equal; and
- exterior angles are euqal.

**The name of a regular polygon of**

- 3 sides is equilateral triangle.
- 4 sides is square.
- 6 sides is regular hexagon.

**Question 6.**

Find the angle measure x in the following figures.

**Solution.**

**Question 7.**

**(a)** Find x + y + z

**(b)** Find x + y + z + w.

**Solution.**

### Chapter 3 Understanding Quadrilaterals Exercise 3.2

**Question 1.**

Find x in the following figures.

**Solution.**

We know that the sum of the exterior angles formed by producing the sides of a convex polygon in the same order is equal to 360°. Therefore,

**(a)** x + 125° + 125° = 360°

⇒ x + 250° = 360°

⇒ x = 360° – 250° = 110°

**(b) **x + 90° +60° + 90° + 70° = 360°**
**⇒ x + 310° = 360°

⇒ x = 360° – 310° = 50°

**Question 2.**

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

**(i)** 9 sides

**(ii)** 15 sides

**Solution.**

**(i)** Each exterior angle of a regular polygon of 9 sides

**(ii)** Each exterior angle of a regular polygon of 15 sides

**Question 3.**

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

**Solution.**

Since the number of sides of a regular polygon

**Question 4.**

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

**Solution.**

Let there be n sides of the polygon. Then, its each interior angle

Thus, there are 24 sides of the polygon.

**Question 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 ?

**Solution.**

**(a)** Since the number of sides of a regular polygon

= ,

Which is not a whole number.

A regular polygon with measure of each exterior angle as 22° is not possible.

**(b)** If interior angle = 22°, then its exterior angle = 180° – 22° = 158°.

But 158 does not divide 360 exactly.

Hence, the polygon is not possible.

**Question 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 ?

**Solution.**

**(a)** The equilateral triangle being a regular polygon of 3 sides has the least measure of an interior angle = 60°.

**(b)** Since the minimum interior angle of a regular polygon is equal to 60°, therefore, the maximum exterior angle possible for a regular polygon = 180° – 60° – 120°.

### Chapter 3 Understanding Quadrilaterals Exercise 3.3

**Question 1.**

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

**(i)** AD = ………

**(ii)** ∠DCB = …………….

**(iii)** OC = ……………….

**(iv)** m∠DAB + m∠CDA = …………..

**Solution.**

**(i)** AD = BC : In a parallelogram, opposite sides are euqal.

**(ii)** ∠DCB = ∠DAB : In a parallelogram, opposite angles are equal.

**(iii)** OC = OA : The diagonals of a parallelogram bisect each other.

**(iv)** m∠DAB + m∠CDA = 180° : In a parallelogram, the stun of any two adjacent angles is 180°.

**Question 2.**

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

**Solution.**

**Question 3.**

Can a quadrilateral ABCD be a parallelogram if

**(i)** ∠D + ∠B = 180° ?

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

**(iii)** ∠A = 70° and ∠C = 65°?

**Solution.**

**(i)** If in a quadrilateral ABCD, ∠D + ∠B = 180°, then it is not necessary that ABCD is a parallelogram.

**(ii)** Since AD ≠ BC, i.e., the opposite sides are unequal, so ABCD is not a parallelogram.

**(iii)** Since ∠A ≠ ∠C, i.e., the opposite angles are unequal, so ABCD is not a parallelogram.

**Question 4.**

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

**Solution.**

A rough figure of a quadrilateral ABCD that is not a parallelogram has been drawn with exactly two opposite angles of equal measure such that ∠A – ∠C which is a kite as an example.

**Question 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.

**Solution.**

Let two adjacent angles A and B of ||gm ABCD be 3x and 2x respectively.

Since the adjacent angles of a parallelogram are supplementary.

Since the opposite angles are equal in a parallelogram, therefore,

∠C = ∠A = 108° and ∠D = ∠B = 72°

Hence, ∠A = 108°, ∠B = 72°, ∠C = 108° and ∠D = 72°.

**Question 6.**

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

**Solution.**

Let two adjacent angles A and B of parallelogram ABCD be x each.

Since the adjacent angles of a parallelogram are supplementary.

**Question 7.**

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

**Solution.**

Since HOPE is a parallelogram, therefore, HE || OP and HO || EP.

Now, HE || OP and transversal HO intersects them.

Hence, x = 110°, y = 40° and z = 30°

**Question 8.**

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

**Solution.**

**(i)** Since GUNS is a parallelogram, therefore, its opposite sides are equal.

**(ii)** In a parallelogram, diagonals bisect each other, therefore,

**Question 9.**

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

**Solution.**

**Question 10.**

Explain how this figure is a trapezium. Which of its two sides are parallel ?

**Solution.**

Since ∠KLM + ∠NML = 180° i.e., the pair of consecutive interior angles are supplmentary.

Therefore, KL || NM and so KLMN is a trapezium.

**Question 11.**

Find m∠C in the figure, if || .

**Solution.**

Since AB || DC and transversal BC intersects

them.

∠B + ∠C= 180°

[∵ Sum of interior angles is 180°]

⇒ 120°+ ∠C =180° A

⇒ ∠C = 180°- 120° = 60°

Hence, m∠C =60°

**Question 12.**

Find the measure of ∠P and ∠S, if || in the figure. (If you find mZ R, is there more than one method to find m∠P ?)

**Solution.**

Since SP || RQ and PQ is a transversal intersecting them at P and Q.

∴ ∠P + ∠Q =180°

[∵ Sum of interior angles is 180°]

### Chapter 3 Understanding Quadrilaterals Exercise 3.4

**Question 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.

**Solution.**

**(a)** False

**(b)** True

**(c)** True

**(d)** False

**(e)** False

**(f)** True

**(g)** True

**(h)** True

**Question 2.**

**Identify all the quadrilaterals that have.**

**(a)** four sides of equal length

**(b)** four right angles

**Solution.**

**(a)** The quadrilaterals having four sides of equal length is either a square or a rhombus. ,

**(b)** The quadrilaterals having four right angles is either a square or a rectangle.

**Question 3.**

**Explain how a square is**

**(i)** a quadrilateral

**(ii)**a parallelogram

**(iii)** a rhombus

**(iv)** a rectangle.

**Solution.**

**(i)** A square is 4 sided, so it is a quadrilateral.

**(ii)** A square has its opposite sides parallel, so it is a parallelogram.

**(iii)** A square is a parallelogram with all the four sides equal, so it is a rhombus.

**(iv)** A square is a parallelogram with each angle a right angle, so it is a rectangle.

**Question 4.**

**Name the quadrilaterals whose diagonals :**

**(i)** bisect each other

**(ii)** are perpendicular bisectors of each other

**(iii)** are equal.

**Solution.**

**(i)** The quadrilaterals whose diagonals bisect each other can be a parallelogram or a rhombus or a square or a rectangle.

**(ii)** The quadrilaterals whose diagonals are perpendicular bisectors of each other can be a rhombus or a square.

**(iii)** The quadrilaterals whose diagonals are equal can be a square or a rectangle.

**Question 5.**

Explain why a rectangle is a convex quadrilateral.

**Solution.**

Since the measure of each angle is less than 180° and also both the diagonals of a rectangle he wholly in its interior, so a rectangle is a convex quadrilateral.

**Question 6.**

ABC is a right-angled triangle and O is the mid-point 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).

**Solution.**

Produce BO to D such that BO = OD. Join AD and DC. Then ABCD is a rectangle. In the rectangle ABCD, its diagonals AC and BD are equal and bisect each other at O.

∴ OA = OC and OB = OD.

But AC = BD

Therefore, OA = OB = OD

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

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