Vertically Opposite Angles & Vertical Angles Theorem

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.

Vertically Opposite Angles

In the figure given above, the line segment \(\overline{AB}\) and \(\overline{CD}\) meet at the point \(O\) and these represent two intersecting lines. The line segment \(\overline{PQ}\) and \(\overline{RS}\) represent two parallel lines as they have no common intersection point in the given plane. Infinite lines can pass through a single point and hence through \(O\), multiple intersecting lines can be drawn. Similarly, infinite parallel lines can be drawn parallel to \(\overline{PQ}\) and \(\overline{RS}\) . Also, it is worth noting that the perpendicular distance between two parallel lines is constant.

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 \(\overleftrightarrow{AB}\) and \(\overleftrightarrow{CD}\) which intersect each other at \(O\). The two pairs of vertical angles are: i) ∠AOD and ∠COB ii) ∠AOC and ∠BOD as shown.

Vertically Opposite Angles

It can be seen that ray \(\overline{OA}\) stands on the line \(\overleftrightarrow{CD}\) and according to Linear Pair Axiom, if a ray stands on a line, then the adjacent angles form a linear pair of angles.

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

Similarly, \(\overline{OC}\) stands on the line \(\overleftrightarrow{AB}\).

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

From (1) and (2),

∠AOD + ∠AOC = ∠AOC + ∠BOC

⇒ ∠AOD = ∠BOC —(3)

Also, \(\overline{OD}\) stands on the line \(\overleftrightarrow{AB}\)..

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.

Vertically Opposite Angles

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

⇒ ∠BOD = 105° and ∠AOD = 75°

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Practise This Question

A ray OC is constructed on the line segment AB such that ∠BOC is x. What will ∠AOC be?