# Vertically Opposite Angle

When two lines intersect, then the vertically opposite angles are equal.

The angles opposite to each other around same vertex are called Vertically Opposite Angles.

⟹ These angles are equal in measurement
⟹ Are also called Vertically Angles

Observe the below image:

Here, \angle 1\ & \angle 2\ are vertically opposite angles.
Note both angle 1 & 2 have common vertex O and are opposite to each other.

Similarly, \angle 3\ & \angle 4\ are vertically opposite angles

Hence,
\angle 1\ =\ \angle 2
\angle 3\ =\ \angle 4

Examples of Vertically Opposite Angles

In the above example, there are four pair of vertically angles;
\angle 1\ =\ \angle 3
\angle 2\ =\ \angle 4
\angle 5\ =\ \angle 7
\angle 6\ =\ \angle 8

## Vertical Angle Theorem

Theorem
When two lines intersect, the vertically opposite angles are equal

Given:
Two lines M & N intersect at point O and four angles are formed as shown in the below image

To Prove:
Vertically Opposite Angles 1 & 2 are equal
i.e. \angle 1\ =\ \angle 2

Solution
We know that N is a straight line.
So angle formed in a straight line will add to 180 degree
\angle 1\ +\ \angle 3 = 180 —– eq(1)

Similarly M is a straight line
\angle 3\ +\ \angle 2 = 180 —– eq(2)

From eq(1) & eq(2), we get:
\angle 1\ +\ \angle 3 = \angle 3\ +\ \angle 2

\angle 1\ =\ \angle 2

Hence Proved

## Frequently Asked Questions – Vertically Angles

(01) Is supplementary angle and Vertically angles same?

No!!
If sum of angle pair = 180, then the angle is known as supplementary

While Vertically opposite angles are just angles that lie opposite to each other through common vertex.
Also, Vertically angles are equal to each other.

(02) Can we have Vertically Opposite angles on Parallel lines?

No!
Parallel lines do not intersect.

In order to form vertical angles, its necessary that two lines intersect each other.

(03) Can the sum of vertical angle pair be 180 degree?

Yes!!
When lines are perpendicular to each other, then the sum of vertical angle can be 180 degree

Observe the above perpendicular lines.
Here angle A & B are vertically opposite angles
\angle a\ =\ \angle b = 90 degree

So if we add both the angles we get:
\angle a\ +\ \angle b = 180 degree

## Vertically Opposite Angles Problem

(01) Find the value of angle x and y

Solution
Here ∠x = ∠A are vertically opposite angles.

So, ∠ x = ∠ A = 110 degree

Similarly,
∠ y = ∠ B = 70 degree [Vertically opposite angle ]

(02) Find angle C in the below figure

Solution
If you observe carefully, you will find that:

∠ a = ∠ MOP – – – – -eq(1)

But MOP is divided into two angles:
∠ MOP = ∠ b + ∠ c – – – eq(2)

Putting eq(2) in eq(1), we get:
∠a = ∠ b + ∠ c

95 = 37 + ∠ c

∠ c = 95 – 37

∠ c = 58 degree

Hence, value of ∠c is 58 degree.

(03) Find angle y in the below figure

You can see that line M is a straight line.

\angle a\ + \angle x\ + \angle b\ = 180

40 + \angle x\ + 31 = 180

\angle x\ = 180 -71

\angle x\ = 109 degree

You can observe that Angle x & y are vertically opposite angles
So, \angle y\ = \angle x\

\angle y\ = 109 degree