CAT Vertically Opposite Angles - Practice Questions & MCQ

Vertically Opposite Angles: When two lines intersect, then the angles having their arms in the opposite direction are called vertically opposite angles. The pair of vertically opposite angles is equal. 

One pair is ∠AOD and ∠BOC. and another pair is ∠AOC and ∠BOD.

Theorem 1 : If two lines intersect each other, then the vertically opposite angles are equal.

Let two lines AB and CD intersect at a point O. Then, two pairs of vertically opposite angles are formed:

(i) ∠AOC and ∠BOD  (ii) ∠AOD and ∠BOC.

We need to prove that ∠AOC = ∠BOD and ∠AOD = ∠BOC..

Since ray OA stands on line CD, we have

Therefore,        ∠AOC + ∠AOD = 180°           [linear pair].

Again, ray OD stands on line AB

So,                  ∠AOD + ∠BOD = 180°           [linear pair].

From above 2 equation, we can write

                      ∠AOC + ∠AOD = ∠AOD + ∠BOD

his implies that            ∠AOC = ∠BOD

Similarly, it can be proved that ∠AOD = ∠BOC.

Remark:

The sum of all the angles formed on the same side of a line at a given point on the line is 180°.

The sum of all angles around a point is 360°.