Vertically Opposite Angles Theorem

Vertical angles theorem or Vertically opposite angles theorem states that,

If two lines intersect each other, then vertically opposite angles are equal.

Given: In the above statement, given that two lines intersect each other, so let

AB and CD are two lines intersect each other at O as shown in figure so that,

two pair of vertical angles are

a. ∠AOC  and ∠BOD
b. ∠AOD  and ∠BOC

We need to prove that angles,

a. ∠AOC = ∠BOD
b. ∠AOD = ∠BOC       

Proof: Ray OA stands on line CD.

Therefore, a.  ∠AOC  + ∠AOD = 180⁰…(1)

∠AOC and ∠AOD are (Linear pair of angles)

Similarly, ∠AOD and ∠BOD are (Linear pair of angles), so that

  ∠AOD  + ∠BOD = 180⁰…(2)

From (1) and (2) we write

∠AOC  + ∠AOD = ∠AOD  + ∠BOD

∠AOC = ∠AOD  + ∠BOD – ∠AOD

∠AOC =  ∠BOD  

Similarly we can prove that,

  ∠AOD  = ∠BOC

                                                    Hence proved