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
\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
When two lines intersect, the vertically opposite angles are equal
Two lines M & N intersect at point O and four angles are formed as shown in the below image
Vertically Opposite Angles 1 & 2 are equal
i.e. \angle 1\ =\ \angle 2
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
Frequently Asked Questions – Vertically Angles
(01) Is supplementary angle and Vertically angles same?
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?
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?
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
Here ∠x = ∠A are vertically opposite angles.
So, ∠ x = ∠ A = 110 degree
∠ y = ∠ B = 70 degree [Vertically opposite angle ]
(02) Find angle C in the below figure
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.
So all the angle made will add up to 180 degree
\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