Vertical Angles:

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. The given figure shows intersecting and parallel lines.

In the figure given above, the line segment \(\overline{AB}\)

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. These angles are also known as vertical angles or opposite angles.

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

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

Proof: Consider two lines \(\overleftrightarrow{AB}\)

It can be seen that ray \(\overline{OA}\)

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

Similarly, \(\overline{OC}\)

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

From (1) and (2),

∠AOD + ∠AOC = ∠AOC + ∠BOC

⇒ ∠AOD = ∠BOC —(3)

Also, \(\overline{OD}\)

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.

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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