Pairs of Angles – Lines & Angles
Last Updated :
30 Oct, 2024
When two lines share a common endpoint, called Vertex then an angle is formed between these two lines and when these angles appear in groups of two to display a specific geometrical property then they are called pairs of angles. Understanding these angle pairs helps in solving geometry problems involving parallel lines, transversals, and polygons.
Below is the pictorial representation of the pair of angles.
Here the angles ∠ABC and ∠ACB can be called a pair of angles.
Some of the angle pairs that we see in geometry are Complementary angles, Supplementary angles, Adjacent angles, Linear pair of angles, and Vertical Angles.
In this article we will discuss about pair of angles and their types based on their properties.
Complementary Angles
When we have two angles whose addition equals 90° then the angles are called Complementary Angles.
Example:
50° and 40° (50° + 40° = 90°)
70° and 20° (70° + 20° = 90°)
Below is the pictorial representation of the Complementary Angles.
- If we have two angles as x° and y° and x° + y° = 90° then x is called the complementary angle of y and y is called the complementary angle of x.
Example: We have 20° and 70° then, 20° is a complementary angle of 70° and 70° is a complementary angle of 20°.
- If we have one angle as x° then to find a complementary angle we need to subtract it from 90°.
Example: We have 30° then the complementary angle of it is 90° – 30° which is 60°
Supplementary Angles
When we have two angles whose addition equals to 180° then the angles are called Supplementary Angles.
Example:
150° and 30° (150° + 30° = 180°)
70° and 110° (70° + 110° = 180°)
Below is the pictorial representation of the Supplementary Angle.
- If we have two angles as x° and y° and x° + y° = 180° then x is called the supplementary angle of y and y is called the supplementary angle of x.
Example: We have 100° and 80° then, 100° is the supplementary angle of 80° and 80° supplementary angle of 100°.
- If we have one angle as x° then to find a supplementary angle we need to subtract it from 180°.
Example: We have 60° then the supplementary angle of it is 180° – 60° which is 120°
Difference Between Complementary Angle and Supplementary Angle
Complementary Angle
|
Supplementary Angle
|
| Sum of both the angle equals to 90°. |
Sum of both the angle equals to 180°. |
| Both the angles are called complements of each other. |
Both the angles are called supplement of each other. |
| If one angle is x°, its complement is 90° – x°. |
If one angle is x°, its supplement is 180° – x°. |
Adjacent Angles
When we have two angles with a common side, a common vertex without any overlap we call them Adjacent Angles.
We know what conditions two angles need to fulfill to be Adjacent angles. Let’s see some of the examples where we might get confused that whether they are adjacent angles or not.
Here θ1 and θ2 are having a common vertex, they don’t overlap but because they don’t share any common side they aren’t Adjacent Angles.
Here θ1 and θ2 are having a common vertex, they share a common side but they overlap so they aren’t Adjacent Angles.
Linear Pair of Angles
We say two angles as linear pairs of angles if both the angles are adjacent angles with an additional condition that their non-common side makes a Straight Line.
Let’s see some examples for a better understanding of Pair of Angles.
Example 1:
Let’s call the intersection of line AC and BD to be O. Now we see four angles are there let’s try to observe them one by one.
- θ1 and θ2 are adjacent angles and their non-common sides are AO and OC, AO + OC = AC is a Straight Line so both are linear pairs of angles.
- θ2 and θ3 are adjacent angles and their non-common sides are BO and OD, BO + OD = BD is a Straight Line so both are linear pairs of angles.
- θ3 and θ4 are adjacent angles and their non-common sides are CO and OA, CO + OA = CA is a Straight Line so both are linear pairs of angles.
- θ4 and θ1 are adjacent angles and their non-common sides are D0 and OB, DO + OB = DB is a Straight Line so both are linear pair of angles.
Vertical Angles
A vertical angle is a pair of non-adjacent angles that are formed by the intersection of two Straight Lines.
Here we see line AD and line BC intersect at one point let’s call it X and thus four angles are formed
∠AXB = θ1
∠BXD = θ2
∠DXC = θ3
∠CXA = θ4
θ1 and θ2 are non-adjacent angles and formed by the intersection of line AD and BC therefore they are Vertical Angles are always Equal so θ1 = θ2. Similarly, θ3 and θ4 are also vertical angles therefore θ3 = θ4. Let’s try to understand with a question:
Here we see ∠BXD and b are vertically opposite angles therefore
b = ∠BXD
b = 60°
and we also see that ∠DXC and a are vertically opposite angles therefore
a = ∠DXC
a = 120°
Example Problems on Pair of Angles
Question 1: Two adjacent angles are formed when a straight line intersects another line. One angle measures 65°. What is the measure of the other adjacent angle?
Solution:
Given:
One angle is 65°.
The two angles are adjacent and form a straight line (sum of 180°).
Then,
By subtracting the given angle from 180°
180°−65°=115°
The other adjacent angle is 115°.
Question 2: Two angles are complementary, and one angle measures 40°. Find the measure of the other angle.
Solution :
Given:
One angle is 40°.
The two angles are complementary (sum of 90°).
then,
By subtracting the given angle form 90°
90°−40°=50°.
The other complementary angle is 50°.
Question 3: Two angles are supplementary. One angle measures 120°. What is the measure of the other angle?
Solution:
Given:
One angle is 120°.
The two angles are supplementary (sum of 180°).
then,
By subtracting 120° from 180°:
180°−120°=60°.
The other supplementary angle is 60°.
Question 4: Two intersecting lines form vertical angles. If one of the angles is 75°, what is the measure of the opposite vertical angle?
Solution:
Given:
One angle is 75°.
then,
As the angles are vertical, meaning they are equal.
The opposite vertical angle is also 75°.
Practices Questions on Pairs of Angles
Question 1. Two angles are complementary. If one angle is 35°, what is the measure of the other angle?
Question 2. Two angles are supplementary. If one angle is 120°, what is the measure of the other angle?
Question 3. If two lines intersect and form a pair of vertical angles, and one angle measures 75°, what is the measure of the other vertical angle?
Question 4. Two adjacent angles form a straight line. If one angle measures 50°, what is the measure of the adjacent angle?
Question 5. Two parallel lines are cut by a transversal. If one of the alternate interior angles is 110°, what is the measure of the other alternate interior angle?
Question 6. Two parallel lines are cut by a transversal. If one of the corresponding angles measures 60°, what is the measure of the other corresponding angle?
Question 7. If two angles form a linear pair and one angle measures 90°, what is the measure of the other angle?
Question 8. Find the measure of an angle that is supplementary to a 30° angle and complementary to a 50° angle.
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