RD Sharma Solution for Class 9 Chapter 8 includes several exercises of Lines and Angles to help the students practice the concepts more effectively. Students can now freely access RD Sharma Class 9 Maths solutions for chapter 8 here. The chapter deals with lines and angles, its different types and formulas etc. Understanding this chapter is quite crucial and students will have to put in some amount of time to practice all questions to get fully acquainted with the chapter.

Students can download RD Sharma Solutions Class 9 for all chapters solutions enlisted in the textbook.

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__Exercise 8.1__

**Question 1: Write the complement of each of the following angles:**

** (i)20 ^{0} **

**(ii)35 ^{0}**

**(iii)90 ^{0} **

**(iv) 77 ^{0} **

**(v)30 ^{0}**

**Solution**:

**(i)** The sum of an angle and its complement = 90^{0}

Therefore, the complement of 20** ^{0 }**= 90

**– 20**

^{0}**= 70**

^{0}

^{0}**(ii)** The sum of an angle and its complement = 90^{0}

Therefore, the complement of 35° = 90° – 35° = 55

**(iii)** The sum of an angle and its complement = 90^{0}

Therefore, the complement of 90** ^{0}** = 90

**– 90**

^{0}**= 0**

^{0}

^{0}**(iv)** The sum of an angle and its complement = 90^{0}

Therefore, the complement of 77** ^{0}** = 90° – 77

**= 13**

^{0}

^{0}**(v)** The sum of an angle and its complement = 90^{0}

Therefore, the complement of 30** ^{0}** = 90

**– 30**

^{0}**= 60**

^{0}

^{0}**Question 2 : Write the supplement of each of the following angles:**

** (i) 54 ^{0}**

**(ii) 132 ^{0}**

** (iii) 138 ^{0}**

**Solution:**

**(i)** The sum of an angle and its supplement = 180** ^{0}**.

Therefore supplement of angle 54** ^{0}** = 180

**– 54**

^{0}**= 126**

^{0}

^{0}**(ii)** The sum of an angle and its supplement = 180** ^{0}**.

Therefore supplement of angle 132** ^{0}** = 180

**– 132**

^{0}**= 48**

^{0}

^{0}**(iii)** The sum of an angle and its supplement = 180** ^{0}**.

Therefore supplement of angle 138** ^{0 }**= 180

**– 138**

^{0 }**= 42**

^{0}

^{0}**Question 3: If an angle is 28 ^{0} less than its complement, find its measure?**

**Solution:**

Let the measure of any angle is ‘ a ‘ degrees

Thus, its complement will be (90 – a)^{ 0}

So, the required angle = Complement of a – 28

a = ( 90 – a ) – 28

2a = 62

a = 31

Hence, the angle measured is 31** ^{0}**.

**Question 4 : If an angle is 30° more than one half of its complement, find the measure of the angle?**

**Solution**:

Let an angle measured by ‘ a ‘ in degrees

Thus, its complement will be (90 – a)^{ 0}

Required Angle = 30** ^{0}** + complement/2

a = 30** ^{0}** + ( 90 – a )

**/ 2**

^{ 0}a + a/2 = 30** ^{0}** + 45

^{0}3a/2 = 75^{0}

a = 50^{0}

Therefore, the measure of required angle is 50** ^{0}**.

**Question 5 : Two supplementary angles are in the ratio 4:5. Find the angles?**

**Solution**:

Two supplementary angles are in the ratio 4:5.

Let us say, the angles are 4a and 5a (in degrees)

Since angle are supplementary angles;

Which implies, 4a + 5a = 180^{0}

9a = 180^{0}

a = 20^{0}

Therefore, 4a = 4 (20) = 80** ^{0 }**and

5(a) = 5 (20) = 100^{0}

Hence, required angles are 80° and 100** ^{0}**.

**Question 6 : Two supplementary angles differ by 48 ^{0}. Find the angles?**

**Solution**: Given: Two supplementary angles differ by 48** ^{0}**.

Consider a** ^{0}** be one angle then its supplementary angle will be equal to (180 – a)

^{ 0}According to the question;

(180 – a ) – x = 48

(180 – 48 ) = 2a

132 = 2a

132/2 = a

Or a = 66^{0}

Therefore, 180 – a = 114^{0}

Hence, the two angles are 66** ^{0}** and 114

**.**

^{0}**Question 7: An angle is equal to 8 times its complement. Determine its measure?**

**Solution:** Given: Required angle = 8 times of its complement

Consider a** ^{0}** be one angle then its complementary angle will be equal to (90 – a)

^{ 0}According to the question;

a = 8 times of its complement

a = 8 ( 90 – a )

a = 720 – 8a

a + 8a = 720

9a = 720

a = 80

Therefore, the required angle is 80** ^{0}**.

__Exercise 8.2__

**Question 1: In the below Fig. OA and OB are opposite rays:**

** (i) If x = 25 ^{0}, what is the value of y?**

**(ii) If y = 35 ^{0}, what is the value of x?**

** **

**Solution:**

**(i)** Given: x = 25

From figure: ∠AOC and ∠BOC form a linear pair

Which implies, ∠AOC + ∠BOC = 180^{0}

From the figure, ∠AOC = 2y + 5 and ∠BOC = 3x

∠AOC + ∠BOC = 180^{0}

(2y + 5) + 3x = 180

(2y + 5) + 3 (25) = 180

2y + 5 + 75 = 180

2y + 80 = 180

2y = 100

y = 100/2 = 50

Therefore, y = 50^{0}_{ }

**(ii)** Given: y = 35^{0}

From figure: ∠AOC + ∠BOC = 180° (Linear pair angles)

(2y + 5) + 3x = 180

(2(35) + 5) + 3x = 180

75 + 3x = 180

3x = 105

x = 35

Therefore, x = 35^{0}

**Question 2: In the below figure, write all pairs of adjacent angles and all the linear pairs.**

** **

**Solution**: From figure, pairs of adjacent angles are :

(∠AOC, ∠COB) ; (∠AOD, ∠BOD) ; (∠AOD, ∠COD) ; (∠BOC, ∠COD)

And Linear pair of angles are (∠AOD, ∠BOD) and (∠AOC, ∠BOC).

[As ∠AOD + ∠BOD = 180^{0}and ∠AOC+ ∠BOC = 180

^{0}.]

**Question 3 : In the given figure, find x. Further find ∠BOC , ∠COD and ∠AOD.**

**Solution:**

From figure, ∠AOD and ∠BOD form a linear pair,

Therefore, ∠AOD+ ∠BOD = 180^{0}

Also, ∠AOD + ∠BOC + ∠COD = 180^{0}

Given: ∠AOD = (x+10)^{ 0} , ∠COD = x^{0} and ∠BOC = (x + 20)^{ 0}

( x + 10 ) + x + ( x + 20 ) = 180

3x + 30 = 180

3x = 180 – 30

x = 150/3

x = 50^{0}

Now,

∠AOD=(x+10) =50 + 10 = 60

∠COD = x = 50

∠BOC = (x+20) = 50 + 20 = 70

Hence, ∠AOD=60^{0}, ∠COD=50^{0} and ∠BOC=70^{0}

**Question 4: In figure, rays OA, OB, OC, OD and OE have the common end point 0. Show that ∠AOB+∠BOC+∠COD+∠DOE+∠EOA=360°.**

**Solution:**

Given: Rays OA, OB, OC, OD and OE have the common endpoint O.

Draw an opposite ray OX to ray OA, which make a straight line AX.

From figure:

∠AOB and ∠BOX are linear pair angles, therefore,

∠AOB +∠BOX = 180^{0}

Or, ∠AOB + ∠BOC + ∠COX = 180^{0} —–—–(1)

Also,

∠AOE and ∠EOX are linear pair angles, therefore,

∠AOE+∠EOX =180°

Or, ∠AOE + ∠DOE + ∠DOX = 180^{0} —–(2)

By adding equations, (1) and (2), we get;

∠AOB + ∠BOC + ∠COF + ∠AOE + ∠DOE + ∠DOX = 180^{0} + 180^{0}

∠AOB + ∠BOC + ∠COD + ∠DOE + ∠EOA = 360^{0}

Hence Proved.

**Question 5 : In figure, ∠AOC and ∠BOC form a linear pair. If a – 2b = 30°, find a and b?**

**Solution:**

Given : ∠AOC and ∠BOC form a linear pair.

=> a + b = 180** ^{0 }** …..(1)

a – 2b = 30^{0} …(2) (given)

On subtracting equation (2) from (1), we get

a + b – a + 2b = 180 – 30

3b = 150

b = 150/3

b = 50^{0}

Since, a – 2b = 30^{0}

a – 2(50) = 30

a = 30 + 100

a = 130^{0}

Therefore, the values of a and b are 130° and 50° respectively.

**Question 6: How many pairs of adjacent angles are formed when two lines intersect at a point?**

**Solution**: Four pairs of adjacent angles are formed when two lines intersect each other at a single point.

For example, Let two lines AB and CD intersect at point O.

The 4 pair of adjacent angles are :

(∠AOD,∠DOB),(∠DOB,∠BOC),(∠COA, ∠AOD) and (∠BOC,∠COA).

**Question 7: How many pairs of adjacent angles, in all, can you name in figure given?**

**Solution**: Number of Pairs of adjacent angles, from the figure, are :

∠EOC and ∠DOC

∠EOD and ∠DOB

∠DOC and ∠COB

∠EOD and ∠DOA

∠DOC and ∠COA

∠BOC and ∠BOA

∠BOA and ∠BOD

∠BOA and ∠BOE

∠EOC and ∠COA

∠EOC and ∠COB

Hence, there are 10 pairs of adjacent angles.

**Question 8: In figure, determine the value of x.**

**Solution**:

The sum of all the angles around a point O is equal to 360°.

Therefore,

3x + 3x + 150 + x = 360^{0}

7x = 360^{0} – 150^{0}

7x = 210^{0}

x = 210/7

x = 30^{0}

Hence, the value of x is 30°.

**Question 9: In figure, AOC is a line, find x.**

**Solution:**

From the figure, ∠AOB and ∠BOC are linear pairs,

∠AOB +∠BOC =180°

70 + 2x = 180

2x = 180 – 70

2x = 110

x = 110/2

x = 55

Therefore, the value of x is 55^{0}.

**Question 10: In figure, POS is a line, find x.**

**Solution:**

From figure, ∠POQ and ∠QOS are linear pairs.

Therefore,

∠POQ + ∠QOS=180^{0}

∠POQ + ∠QOR+∠SOR=180^{0}

60^{0} + 4x +40^{0} = 180^{0}

4x = 180^{0} -100^{0}

4x = 80^{0}

x = 20^{0}

Hence, the value of x is 20^{0}.

__Exercise 8.3__

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

**Solution**:

Given: x = 45^{0}

Since vertically opposite angles are equal, therefore z = x = 45^{0}

z and u are angles that are a linear pair, therefore, z + u = 180^{0}

Solve, z + u = 180^{0} , for u

u = 180^{0} – z

u = 180^{0} – 45

u = 135^{0}

Again, x and y angles are a linear pair.

x+ y = 180^{0}

y = 180^{0} – x

y =180^{0} – 45^{0}

y = 135^{0 }

Hence, remaining angles are y = 135^{0}, u = 135^{0} and z = 45^{0}.

**Question 2 : In figure, three coplanar lines intersect at a point O, forming angles as shown in the figure. Find the values of x, y, z and u .**

**Solution**:

(∠BOD, z); (∠DOF, y ) are pair of vertically opposite angles.

So, ∠BOD = z = 90^{0}

∠DOF = y = 50^{0}

Now, x + y + z = 180 [Linear pair] [AB is a straight line]

x + y + z = 180

x + 50 + 90 = 180

x = 180 – 140

x = 40

Hence values of x, y, z and u are 40^{0}, 50^{0}, 90^{0} and 40^{0} respectively.

**Question 3 : In figure, find the values of x, y and z.**

**Solution:**

From figure,

y = 25^{0} [Vertically opposite angles are equal]

Now ∠x + ∠y = 180^{0} [Linear pair of angles]

x = 180 – 25

x = 155

Also, z = x = 155 [Vertically opposite angles]

Answer: y = 25^{0} and z = 155^{0}

**Question 4 : In figure, find the value of x.**

**Solution:**

∠AOE = ∠BOF = 5x [Vertically opposite angles]

∠COA+∠AOE+∠EOD = 180^{0} [Linear pair]

3x + 5x + 2x = 180

10x = 180

x = 180/10

x = 18

The value of x = 18^{0}

**Question 5 : Prove that bisectors of a pair of vertically opposite angles are in the same straight line.**

**Solution:**

Lines AB and CD intersect at point O, such that

∠AOC = ∠BOD (vertically angles) …(1)

Also OP is the bisector of AOC and OQ is the bisector of BOD

To Prove: POQ is a straight line.

OP is the bisector of ∠AOC:

∠AOP = ∠COP …(2)

OQ is the bisector of ∠BOD:

∠BOQ = ∠QOD …(3)

Now,

Sum of the angles around a point is 360^{o}.

∠AOC + ∠BOD + ∠AOP + ∠COP + ∠BOQ + ∠QOD = 360^{0}

∠BOQ + ∠QOD + ∠DOA + ∠AOP + ∠POC + ∠COB = 360^{0}

2∠QOD + 2∠DOA + 2∠AOP = 360^{0} (Using (1), (2) and (3))

∠QOD + ∠DOA + ∠AOP = 180^{0}

POQ = 180^{0}

Which shows that, the bisectors of pair of vertically opposite angles are on the same straight line.

Hence Proved.

**Question 6 : If two straight lines intersect each other, prove that the ray opposite to the bisector of one of the angles thus formed bisects the vertically opposite angle.**

**Solution:** Given AB and CD are straight lines which intersect at O.

OP is the bisector of ∠ AOC.

To Prove : OQ is the bisector of ∠BOD

Proof :

AB, CD and PQ are straight lines which intersect in O.

Vertically opposite angles: ∠ AOP = ∠ BOQ

Vertically opposite angles: ∠ COP = ∠ DOQ

OP is the bisector of ∠ AOC : ∠ AOP = ∠ COP

Therefore, ∠BOQ = ∠ DOQ

Hence, OQ is the bisector of ∠BOD.

__Exercise 8.4__

**Question 1: In figure, AB, CD and ∠1 and ∠2 are in the ratio 3 : 2. Determine all angles from 1 to 8.**

**Solution: **

Let ∠1 = 3x and ∠2 = 2x

From figure: ∠1 and ∠2 are linear pair of angles

Therefore, ∠1 + ∠2 = 180

3x + 2x = 180

5x = 180

x = 180 / 5

=> x = 36

So, ∠1 = 3x = 108^{0} and ∠2 = 2x = 72^{0}

As we know, vertically opposite angles are equal.

Pairs of vertically opposite angles are:

(∠1 = ∠3); (∠2 = ∠4) ; (∠5, ∠7) and (∠6 , ∠8)

∠1 = ∠3 = 108°

∠2 = ∠4 = 72°

∠5 = ∠7

∠6 = ∠8

We also know, if a transversal intersects any parallel lines, then the corresponding angles are equal

∠1 = ∠5 = ∠7 = 108°

∠2 = ∠6 = ∠8 = 72°

Answer: ∠1 = 108°, ∠2 = 72°, ∠3 = 108°, ∠4 = 72°, ∠5 = 108°, ∠6 = 72°, ∠7 = 108° and ∠8 = 72°

**Question 2: In figure, I, m and n are parallel lines intersected by transversal p at X, Y and Z respectively. Find ∠1, ∠2 and ∠3.**

**Solution:** From figure:

∠Y = 120° [Vertical opposite angles]

∠3 + ∠Y = 180° [Linear pair angles theorem]

=> ∠3= 180 – 120

=> ∠3= 60°

Line l is parallel to line m,

∠1 = ∠3 [ Corresponding angles]

∠1 = 60°

Also, line m is parallel to line n,

∠2 = ∠Y [Alternate interior angles are equal]

∠2 = 120°

Answer: ∠1 = 60°, ∠2 = 120° and ∠3 = 60°.

**Question 3: In figure, AB || CD || EF and GH || KL. Find ∠HKL.**

**Solution:**

Extend LK to meet line GF at point P.

From figure, CD || GF, so, alternate angles are equal.

∠CHG =∠HGP = 60°

∠HGP =∠KPF = 60° [Corresponding angles of parallel lines are equal]

Hence, ∠KPG =180 – 60 = 120°

=> ∠GPK = ∠AKL= 120° [Corresponding angles of parallel lines are equal]

∠AKH = ∠KHD = 25° [alternate angles of parallel lines]

Therefore, ∠HKL = ∠AKH + ∠AKL = 25 + 120 = 145°

**Question 4: In figure, show that AB || EF.**

**Solution: **Produce EF to intersect AC at point N.

From figure, ∠BAC = 57° and

∠ACD = 22°+35° = 57°

Alternative angles of parallel lines are equal

=> BA || EF …..(1)

Sum of Co-interior angles of parallel lines is 180°

EF || CD

∠DCE + ∠CEF = 35 + 145 = 180° …(2)

From (1) and (2)

AB || EF

[Since, Lines parallel to the same line are parallel to each other]Hence Proved.

**Question 5 : In figure, if AB || CD and CD || EF, find ∠ACE.**

**Solution: **

Given: CD || EF

∠ FEC + ∠ECD = 180°

[Sum of co-interior angles is supplementary to each other]=> ∠ECD = 180° – 130° = 50°

Also, BA || CD

=> ∠BAC = ∠ACD = 70°

[Alternative angles of parallel lines are equal]But, ∠ACE + ∠ECD =70°

=> ∠ACE = 70° — 50° = 20°

** Question 6: In figure, PQ || AB and PR || BC. If ∠QPR = 102°, determine ∠ABC. Give reasons.**

**Solution:** Extend line AB to meet line PR at point G.

Given: PQ || AB,

∠QPR = ∠BGR =102°

[Corresponding angles of parallel lines are equal]And PR || BC,

∠RGB+ ∠CBG =180°

[Corresponding angles are supplementary]∠CBG = 180° – 102° = 78°

Since, ∠CBG = ∠ABC

=>∠ABC = 78°

**Question 7 : In figure, state which lines are parallel and why?**

**Solution: **

We know, If a transversal intersects two lines such that a pair of alternate interior angles are equal, then the two lines are parallel

From figure:

=> ∠EDC = ∠DCA = 100°

Lines DE and AC are intersected by a transversal DC such that the pair of alternate angles are equal.

So, DE || AC

**Question 8: In figure, if l||m, n || p and ∠1 = 85°, find ∠2.**

**Solution:**

Given: ∠1 = 85°

As we know, when a line cuts the parallel lines, the pair of alternate interior angles are equal.

=> ∠1 = ∠3 = 85°

Again, co-interior angles are supplementary, so

∠2 + ∠3 = 180°

∠2 + 55° =180°

∠2 = 180° – 85°

∠2 = 95°

**Question 9 : If two straight lines are perpendicular to the same line, prove that they are parallel to each other.**

**Solution: **

Let lines l and m are perpendicular to n, then

∠1= ∠2=90°

Since, lines l and m cut by a transversal line n and the corresponding angles are equal, which shows that, line l is parallel to line m.

**Question 10: 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.**

**Solution:** Let the angles be ∠ACB and ∠ABD

Let AC perpendicular to AB, and CD is perpendicular to BD.

To Prove : ∠ACD = ∠ABD OR ∠ACD + ∠ABD =180°

Proof :

In a quadrilateral,

∠A+ ∠C+ ∠D+ ∠B = 360°

[ Sum of angles of quadrilateral is 360° ]=> 180° + ∠C + ∠B = 360°

=> ∠C + ∠B = 360° –180°

Therefore, ∠ACD + ∠ABD = 180°

And ∠ABD = ∠ACD = 90°

Hence, angles are equal as well as supplementary.

__Exercise VSAQs__

**Question 1: Define complementary angles.**

**Solution:** When the sum of two angles is 90 degrees, then the angles are known as complementary angles.

**Question 2: Define supplementary angles.**

**Solution:** When the sum of two angles is 180°, then the angles are known as supplementary angles.

**Question 3: Define adjacent angles.**

**Solution:** Two angles are Adjacent when they have a common side and a common vertex.

**Question 4: The complement of an acute angle is _____.**

**Solution**: An acute angle

**Question 5: The supplement of an acute angle is _____.**

**Solution:** An obtuse angle

**Question 6: The supplement of a right angle is _____.**

**Solution:** A right angle

## RD Sharma Solutions for Class 9 Maths Chapter 8 Lines and Angles

In the 8th chapter of Class 9 RD Sharma Solutions, students will study important concepts listed below:

- Introduction to Lines and Angles
- Angles
- Angles Axioms and Theorems
- Some angle relations
- Linear pair of angles
- Vertically opposite angles
- Angles made by a transversal with two lines
- Alternate interior angles
- Consecutive interior angles