RD Sharma Class 9 Maths solutions chapter 8 exercise 8.3 is available here. All exercise questions are solved by subject experts at BYJU’S using a step-by-step problem-solving approach. RD Sharma Solutions class 9 exercise 8.3 is based on some useful relations on pair of angles, especially vertically opposite angles.
Access Answers to Maths RD Sharma Class 9 Chapter 8 Lines and Angles Exercise 8.3 Page number 8.19
Question 1: In figure, lines l1, and l2 intersect at O, forming angles as shown in the figure. If x = 45. Find the values of y, z and u.
Given: x = 450
Since vertically opposite angles are equal, therefore z = x = 450
z and u are angles that are a linear pair, therefore, z + u = 1800
Solve, z + u = 1800 , for u
u = 1800 – z
u = 1800 – 45
u = 1350
Again, x and y angles are a linear pair.
x+ y = 1800
y = 1800 – x
y =1800 – 450
y = 1350
Hence, remaining angles are y = 1350, u = 1350 and z = 450.
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 .
(∠BOD, z); (∠DOF, y ) are pair of vertically opposite angles.
So, ∠BOD = z = 900
∠DOF = y = 500[Vertically opposite angles are equal.]
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 400, 500, 900 and 400 respectively.
Question 3 : In figure, find the values of x, y and z.
y = 250 [Vertically opposite angles are equal]
Now ∠x + ∠y = 1800 [Linear pair of angles]
x = 180 – 25
x = 155
Also, z = x = 155 [Vertically opposite angles]
Answer: y = 250 and z = 1550
Question 4 : In figure, find the value of x.
∠AOE = ∠BOF = 5x [Vertically opposite angles]
∠COA+∠AOE+∠EOD = 1800 [Linear pair]
3x + 5x + 2x = 180
10x = 180
x = 180/10
x = 18
The value of x = 180
Question 5 : Prove that bisectors of a pair of vertically opposite angles are in the same straight line.
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)
Sum of the angles around a point is 360o.
∠AOC + ∠BOD + ∠AOP + ∠COP + ∠BOQ + ∠QOD = 3600
∠BOQ + ∠QOD + ∠DOA + ∠AOP + ∠POC + ∠COB = 3600
2∠QOD + 2∠DOA + 2∠AOP = 3600 (Using (1), (2) and (3))
∠QOD + ∠DOA + ∠AOP = 1800
POQ = 1800
Which shows that, the bisectors of pair of vertically opposite angles are on the same straight line.
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
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.
RD Sharma Solutions for Class 9 Maths Chapter 8 Exercise 8.3
Class 9 Maths Chapter 8 Lines and Angles Exercise 8.3 is based on vertically opposite angles. Students can freely access RD Sharma Solutions chapter 8 exercise solutions and prepare for their exams.