# RD SHARMA Solutions for Class 9 Maths Chapter 10 - Lines and Angles

## Chapter 10 - Lines and Angles Exercise Ex. 10.1

Write the complement of each of the following angles:

(i) 20^{o}

(ii) 35^{o}

(iii) 90^{o}

(iv) 77^{o}

(v) 30^{o}

Write the supplement of each of the following angles:

(i) 54^{o}

(ii) 132^{o}

(iii) 138^{o}

(i) 54°

Since, the sum of an angle and its supplement is 180°

∴Its supplement will be 180° - 54° = 126°.

(ii) 132°

Since, the sum of an angle and its supplement is 180°

∴Its supplement will be 180° - 132° = 48°.

(iii) 138°

Since, the sum of an angle and its supplement is 180°

∴Its supplement will be 180° - 138° = 42°.

If an angle is 28^{o} less than its complement, find its measure.

If an angle is 30^{o} more than one half of its complement, find the measure of the angle.

If the complement of an angle is equal to the supplement of the thrice of it. Find the measure of angle.

Let the measure of the angle be x^{o}.

Its complement will be (90^{o} - x^{o}) and its supplement will be (180^{o} - x^{o}).

Supplement of thrice of the angle = (180^{o} - 3x^{o})

According to the given information:

(90^{o} - x^{o}) = (180^{o} - 3x^{o})

3x - x = 180 - 90

2x = 90

x = 45

Thus, the measure of the angle is 45^{o}.

The measure of the angle is 45^{o}

## Chapter 10 - Lines and Angles Exercise Ex. 10.2

In fig., OA and OB are opposite rays:

(i) If x = 25°, what is the value of y?

(ii) if y = 35°, what is the value of x?

In fig., write all pairs of adjacent angles and all the linear pairs.

In fig., find x. further find ∠BOC, ∠COD and ∠AOD

In fig., rays OA, OB, OC, OD and OE have the common end point O. Show that ∠AOB + ∠BOC + ∠COD + ∠DOE + ∠EOA = 360º

How many pairs of adjacent angles, in all, can you name in fig.

In fig., determine the value of x.

In fig., AOC is a line, find x.

In Fig., POS is a line, find x.

In fig., ACB is a line such that ∠DCA = 5x and ∠DCB = 4x. Find the values of ∠DCA and∠DCB

Give POR = 3x and QOR = 2x + 10, find the value of x for which POQ will be aline.

What value of y would make AOB a line in fig., if ∠AOC = 4y and ∠BOC = (6y + 30)?

In Fig., Lines PQ and RS intersect each other at point O. If ∠POR : ∠ROQ = 5:7, find all the angles.

In Fig. If a greater than b by one third of a right-angle. find the values of a and b.

^{o}

^{o}

^{o }- ∠POS ... (1)

^{o}(As OR ⊥ PQ)

^{o}

^{o}... (2)

## Chapter 10 - Lines and Angles Exercise Ex. 10.3

In fig., lines l_{1} aans l_{2} intersect at O, forming angles as shown in the figure. If x = 45, find the values of y, z and u.

In fig., three coplanar lines intersect at a point O, forming angles as shown in the figure. Find the values of x, y, z and u.

In fig. , find the values of x, y and z.

In Fig., find the value of X.

In fig., rays AB and CD intersect at O.

(i) Determine y when x = 60^{o}

(ii) Determine x when y = 40

In fig., lines AB, CD and EF intersect at O. Find the measures of ∠AOC, ∠COF, ∠DOE and ∠BOF.

In fig., lines AB, and CD intersect at O. If ∠AOC + ∠BOE = 70° and ∠BOD = 40°, find ∠BOE and reflex ∠COE.

Which of the following statements are true (T) and which are false (F)?

(i) Angles forming a linear pair are supplementary.

(ii) If two adjacent angles are equal, then each angle measures 90^{o}.

(iii) Angles forming a linear pair can both be acute angles.

(iv) If angles forming a linear pair are equal, then each of these angles is of measure 90^{o}.

(i) True

(ii) False

(iii) False

(iv) True

Fill in the blanks so as to make the following statements true:

(i) If one angle of a linear pair is acute, then its other angle will be ________.

(ii) A ray stands on a line, then the sum of the two adjacent angles so formed is _______.

(iii) If the sum of two adjacent angles is 180^{o}, then the ______ arms of the two angles are opposite rays.

(i) obtuse.

(ii) 180^{o}

(iii) uncommon

## Chapter 10 - Lines and Angles Exercise Ex. 10.4

In fig., AB ∥ CD and ∠1 and ∠2 are in the ratio 3:2. determine all angles from 1 to 8.

In fig., l, m and n are parallel lines intersected by transversal p at x, y and z respectively. find ∠1, ∠2, ∠3.

In fig., if AB ∥ CD and CD ∥ EF, find ∠ACE.

In fig., state which lines are parallel and why.

In fig. if l ∥ m, n ∥ p and ∠1 = 85°, find ∠2.

If two straight lines are perpendicular to the same line, prove that they are parallel to each other.

*AB*and

*CD*be perpendicuar to line

*MN*.

In fig., ∠1 = 60° and ∠2 = (2/3)^{rd} of a right angle. prove that l ∥ m

In fig., if l ∥ m ∥ n and ∠1 = 60°, find ∠2.

*AB*and

*CD*be perpendicuar to line

*MN*.

In fig., p is transversal to lines m and n, ∠2 = 120° and ∠5 = 60°. Prove that m ∥ n.

In fig., transceral l intersects two lines m and n, ∠4 = 110° and ∠7 = 65°. is m ∥ n ?

Which pair of lines in Fig., are parallel? give reasons.

In fig., AB ∥ CD ∥ EF and GH ∥ KL. Find ∠HKL

In fig., show that AB ∥ EF.

In Fig., PQ ∥ AB and PR BC. IF ∠QPR = 102º, determine ∠ABC. Give reasons.

Prove that if the two arms of an angle are perpendicular to the two arms of another angle, then the angles are either equal or supplementary.

Consider the angles AOB and ACB.

In fig., lines AB and CD are parallel and p is any point as shown in the figure. Show that ∠ABP + ∠CDP = ∠DPB.

In fig., AB ∥ CD and P is any point shown in the figure. Prove that:

∠ABP + ∠BPD + ∠CDP = 360°

In fig., arms BA and BC of ∠ABC are respectively parallel to arms ED and EF of ∠DEF. Prove that ∠ABC = ∠DEF

## Chapter 10 - Lines and Angles Exercise 10.51

One angle is equal to three times its supplement. The measure of the angle is

(a) 130°

(b) 135°

(c) 90°

(d) 120°

In figure, the value of y is

(a) 20^{°}

(b) 30^{°}

(c) 45^{°}

(d) 60^{°}

## Chapter 10 - Lines and Angles Exercise 10.52

In figure, the value of x is

(a) 12

(b) 15

(c) 20

(d) 30

In figure, which of the following statements must be true?

(i) a + b = d + c (ii) a + c + e = 180^{°} (ii) b + f = c + e

(a) (i) only

(b) (ii) only

(c) (iii) only

(d) (ii) and (iii) only

## Chapter 10 - Lines and Angles Exercise 10.53

If two interior angles on the same side of a transversal intersecting two parallel lines are in the ratio 2 : 3, then the measure of the larger angle is

(a) 54^{°}

(b) 120^{°}

(c) 108^{°}

(d) 136^{°}

## Chapter 10 - Lines and Angles Exercise 10.54

## Chapter 10 - Lines and Angles Exercise 10.55

In figure, if lines l and m are parallel, then x =

(a) 20°

(b) 45°

(c) 65°

(d) 85°

## Chapter 10 - Lines and Angles Exercise 10.56

In figure, if lines l and m are parallel lines, then x =

(a) 70^{°}

(b) 100^{°}

(c) 40^{°}

(d) 30^{°}

In figure, if lines l and m are parallel, then the value of x is

(a) 35^{°}

(b) 55^{°}

(c) 65^{°}

(d) 75^{°}

Two complementary angles are such that two times the measure of one is equal to three times the measure of the other. The measure of the smaller angle is

(a) 45°

(b) 30°

(c) 36°

(d) none of these

Correct option (c)

Let one angle be θ

Then, its complementary = 90 - θ

According to question,

2θ = 3(90 - θ)

5θ = 270

θ = 54°

Then, 90 - θ° = 36°

Hence, the smaller angle is 36°.

Hence, correct option is (c).

## Chapter 10 - Lines and Angles Exercise 10.57

In figure, if line segment AB is parallel to the line segment CD, What is the value of y?

(a) 12

(b) 15

(c) 18

(d) 20

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