Prove “if two lines intersect each other, then the vertically opposite angles are equal.”

Let AB and CD be two lines intersecting at O.

This leads to two pairs of vertically opposite angles, namely,
(i) ∠AOC and ∠BOD
(ii) ∠AOD and ∠BOC
We are to prove that
(i) ∠AOC = ∠BOD
and (ii) ∠AOD = ∠BOC
∵ Ray OA stands on line CD Therefore,
∠AOC + ∠AOD = 180°    ...(1)
| Linear Pair Axiom
∵ Ray OD stands on line AB Therefore,
∠AOD + ∠BOD = 180°    ...(2)
| Linear Pair Axiom
From (1) and (2),
∠AOC + ∠AOD = ∠AOD + ∠BOD
⇒    ∠AOC = ∠BOD
Similarly, we can prove that
∠AOD = ∠BOC

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Ray OE bisects ∠AOB and OF is the ray opposite to OE. Show that ∠FOB = ∠FOA.

Rays OA, OB. OC, OD and OE have the common initial point O. Show that ∠AOB + ∠BOC + ∠COD + ∠DOE + ∠EOA = 360°.

In figure, OP bisects ∠AOC, OQ bisects ∠BOC and OP ⊥ OQ. Show that the points A, O and B are collinear.

In figure, if y = 20°, prove that the line AOB is a straight line.

Lines and Angles

Prove “if two lines intersect each other, then the vertically opposite angles are equal.”