RD Chapter 8 Lines and Angles Ex 8.3 Solutions
Question - 1 : - In figure, lines l1, and l2 intersect at O, formingangles as shown in the figure. If x = 45. Find the values of y, z and u.
Answer - 1 : -
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 andz = 450.
Question - 2 : - In figure, threecoplanar lines intersect at a point O, forming angles as shown in the figure.Find the values of x, y, z and u .
Answer - 2 : -
(∠BOD,z); (∠DOF, y) are pair of vertically opposite angles.
So, ∠BOD = z= 900
∠DOF = y= 500
[Vertically oppositeangles 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.
Answer - 3 : -
From figure,
y = 250 [Vertically opposite angles are equal]
Now ∠x + ∠y = 1800 [Linear pair ofangles]
x = 180 – 25
x = 155
Also, z = x = 155 [Vertically opposite angles]
Answer: y = 250 and z = 1550
Question - 4 : - In figure, find thevalue of x.
Answer - 4 : -
∠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 thatbisectors of a pair of vertically opposite angles are in the same straightline.
Answer - 5 : -
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 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 oppositeangles are on the same straight line.
Hence Proved.
If two straightlines intersect each other, prove that the ray opposite to the bisector of oneof the angles thus formed bisects the vertically opposite angle.
Answer - 6 : -
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.