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Solution:
Given: Let us consider AB and CD as the two lines intersecting at a point O where
OE is the ray bisecting∠BOD and OF is the ray bisecting ∠AOC.

To Prove: ∠AOF = ∠COF
Proof: EF is a straight line passing through the point ‘O’ where two opposite rays
are OE and OF.
From the figure we know that ∠AOF and ∠BOE, ∠COF and ∠DOE are vertically
opposite angles
So, it can be written as
∠AOF = ∠BOE and ∠COF = ∠DOE
It is given that ∠BOE = ∠DOE
So, we can write it as ∠AOF = ∠COF
Therefore, it is proved that ∠AOF = ∠COF.

15. In the given figure, three lines AB, CD and EF intersect at a point O such
0
0
that ∠AOE = 35 and ∠BOD = 40 . Find the measure of ∠AOC, ∠BOF, ∠COF
and ∠DOE.

Solution:
0
It is given that ∠BOD = 40
From the figure we know that ∠BOD and ∠AOC are vertically opposite angles
0
∠AOC = ∠BOD = 40
0
It is given that ∠AOE = 35
From the figure we know that ∠BOF and ∠AOE are vertically opposite angles
0
∠AOE = ∠BOF = 35
From the figure we know that AOB is a straight line
So, it can be written as
0
∠AOB = 180
0
∠AOE + ∠EOD + ∠BOD = 180
0
0
0
35 + ∠EOD + 40 = 180
0
0
0
∠EOD = 180 - 35 - 40
0
∠EOD = 105
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