Vertically Opposite Angles & Vertical Angles Theorem

When two lines intersect each other, then the opposite angles, formed due to intersection are called vertical angles or vertically opposite angles. A pair of vertically opposite angles are always equal to each other. Also, a vertical angle and its adjacent angle are supplementary angles, i.e., they add up to 180 degrees. For example, if two lines intersect and make an angle, say X=45°, then its opposite angle is also equal to 45°. And the angle adjacent to angle X will be equal to 180 – 45 = 135°.

When two lines meet at a point in a plane, they are known as intersecting lines. When the lines do not meet at any point in a plane, they are called parallel lines. Learn about Intersecting Lines And Non-intersecting Lines here.

Definition

As we have discussed already in the introduction, the vertical angles are formed when two lines intersect each other at a point. After the intersection of two lines, there are a pair of two vertical angles, which are opposite to each other.

The given figure shows intersecting lines and parallel lines.

In the figure given above, the line segment

In a pair of intersecting lines, the angles which are opposite to each other form a pair of vertically opposite angles. In the figure given above, ∠AOD and ∠COB form a pair of vertically opposite angle and similarly ∠AOC and ∠BOD form such a pair. Therefore,

∠AOD = ∠COB

∠AOC = ∠BOD

For a pair of opposite angles the following theorem, known as vertical angle theorem holds true.

Note: A vertical angle and its adjacent angle is supplementary to each other. It means they add up to 180 degrees

Vertical Angles: Theorem and Proof

Theorem: In a pair of intersecting lines the vertically opposite angles are equal.

Proof: Consider two lines

i) ∠AOD and ∠COB

ii) ∠AOC and ∠BOD

It can be seen that ray

Therefore, ∠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,

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

From (1) and (4),

∠AOD + ∠AOC = ∠AOD + ∠BOD

⇒ ∠AOC = ∠BOD —(5)

Thus, the pair of opposite angles are equal.

Hence, proved.

Solved Example

Consider the figure given below to understand this concept.

In the given figure ∠AOC = ∠BOD and ∠COB = ∠AOD(Vertical Angles)

⇒ ∠BOD = 105° and ∠AOD = 75°

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