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=45Â°, then its opposite angle is also equal to 45Â°. And the angle adjacent to angle X will be equal to 180 – 45 = 135Â°.

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 aboutÂ Intersecting Lines And Non-intersecting LinesÂ 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 \(\overline{AB}\) 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.

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, âˆ AOD and âˆ COB form a pair of vertically opposite angle and similarly âˆ AOC and âˆ BOD form such a pair. Therefore,

âˆ AOD = âˆ COB

âˆ AOC = âˆ BOD

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

**Note:**Â 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 \(\overleftrightarrow{AB}\) and \(\overleftrightarrow{CD}\) which intersect each other at \(O\). The two pairs of vertical angles are:

i) âˆ AOD and âˆ COB

ii) âˆ AOC and âˆ BOD

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, âˆ AOD + âˆ AOC = 180Â° —(1) (Linear pair of angles)

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

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

From (1) and (2),

âˆ AOD + âˆ AOC = âˆ AOC + âˆ BOC

â‡’ âˆ AOD = âˆ BOC —(3)

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

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

From (1) and (4),

âˆ AOD + âˆ AOC = âˆ AOD + âˆ BOD

â‡’ âˆ AOC = âˆ 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 âˆ AOC = âˆ BOD and âˆ COB = âˆ AOD(Vertical Angles)

â‡’ âˆ BOD = 105Â° and âˆ AOD = 75Â°

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## Frequently Asked Questions â€“ FAQs

### What is vertical angles?

### How to measure vertical angles?

### If x=30 degrees is a vertical angle, when two lines intersect, then find all the angles?

Let y is the angle vertically opposite to x, then y = 30 degrees

Now, as we know, vertical angle and its adjacent angle add up to 180 degrees, therefore,

The other two angles are: 180 â€“ 30 = 150 degrees

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