Vertically Opposite Angles

Definition

When two lines meet at a common point in a plane, they are known as intersecting lines, and if they do not meet anywhere on the plane, then they are known as Parallel Lines. Opposite angles formed due to the intersection of two lines are called vertically opposite angles. These angles are of equal measure. One of the vertically opposite angles along with its adjacent angle measures 180^◦

Theorem and Proof:

If two lines are intersecting each other, the vertically opposite angles are of equal measure.

Proof: Two straight lines intersect at point ‘O’ to generate two pairs of vertically opposing angles, as can be seen in the illustration beside AB and CD.:

1. i) ∠AOD and ∠COB
2. ii) ∠AOC and ∠BOD

It can be seen,

∠AOD + ∠AOC = 180° —(1) (Linear pair of angles)

Similarly,

Therefore, ∠AOC + ∠BOC = 180° —(2) (Linear pair of angles)

From (1) and (2),

∠AOD + ∠AOC = ∠AOC + ∠BOC

⇒ ∠AOD = ∠BOC —(3)

Also,

∠AOD + ∠BOD = 180° —(4) (Linear pair of angles)

From (1) and (4),

∠AOD + ∠AOC = ∠AOD + ∠BOD

⇒ ∠AOC = ∠BOD —(5)

Hence, it is proved that vertically opposite angles formed by the intersection of two lines in a plane are equal.

Solved Examples:

Q1) If the measure of angle AOC in the given figure is 75 degrees. What is the measure of angle BOD in degrees?

A: Angle AOC = Angle BOD (Vertically Opposite Angles)

Hence Angle AOC= Angle BOD = 75 degrees.

Q2) If the measure of angle AOC in the given figure is 10 degrees. What is the measure of angle BOD in degrees?

A: Angle AOC = Angle BOD (Vertically Opposite Angles)

Hence Angle AOC= Angle BOD = 10 degrees.