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Pair of Angles – Definition, Diagrams, Types, and Examples

linear pair of angles

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The figure that is formed when two lines meet or intersect is an angle. Angle is one of the basic concepts of geometry. The pairs of angles are nothing but the two angles. Moreover, if there is one common line for two angles, then it is known as an ‘angle pair’. 

Let’s understand what is meant by pair of angles and what are the different types of pairs of angles with their properties.

What is Pair of Angles?

The term ‘pair of angles’ is used for two angles taken together. Angle pairs are called that because they always appear as two angles working together to display some unusual or interesting property. There are eight types of pairs of angles.

  • Complementary Angles
  • Supplementary Angles
  • Linear Pair of Angles
  • Vertically Opposite Angles
  • Alternate Interior Angles
  • Alternate Exterior Angles
  • Corresponding Angles
  • Adjacent Angles
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1. Complementary Angles

An angle pair that adds up to $90^{\circ}$ is called a pair of complementary angles. These pairs do not have to be touching to be complementary. Any two angles that sum to $90^{\circ}$ are complementary angle pairs.

linear pair of angles

In the above figure, $\angle \text{ABC}$ and $\angle \text{DEF}$ are two separate angles with no arm(or side) in common. The sum of these two angles is $36 + 54 = 90^{\circ}$, therefore, $\angle \text{ABC}$ and $\angle \text{DEF}$ forms a pair of complementary angles.

In the case of $\angle \text{LMO}$ and $\angle \text{OMN}$, these two angles share a common arm(or side) $\text{MO}$ and these also form a pair of complementary angles as $60 + 30 = 90^{\circ}$.

2. Supplementary Angles

An angle pair that adds up to $180^{\circ}$ is called a pair of supplementary angles. These pairs do not have to be touching to be supplementary. Any two angles that sum to $180^{\circ}$ are supplementary angle pairs.

Note: When a pair of supplementary angles share a common arm(or side), it is called a linear pair.

linear pair of angles

In the above figure, $\angle \text{ABC}$ and $\angle \text{DEF}$ are two separate angles with no arm(or side) in common. The sum of these two angles is $70 + 110 = 180^{\circ}$, therefore, $\angle \text{ABC}$ and $\angle \text{DEF}$ forms a pair of supplementary angles.

In the case of $\angle \text{MOL}$ and $\angle \text{LON}$, these two angles share a common arm(or side) $\text{LO}$ and these also form a pair of supplementary angles as $65 + 115 = 180^{\circ}$.

3. Linear Pair of Angles

The two angles are said to be a linear pair of angles if both the angles are adjacent angles with an additional condition that their non-common side makes a straight line (an angle of $180^{\circ}$).

linear pair of angles

In the above figure, the angles $\angle \text{ABD}$ and  $\angle \text{DBC}$ form a linear pair as the sum of these two angles is $125 + 55 = 180^{\circ}$ and they share a common arm(or side) $\text{DB}$ and the non-common sides $\text{AB}$ and $\text{BC}$ forms a straight line $\text{AC}$.

In the second figure also, the linear pair of angles are

  • $\angle \text{XOQ}$ and $\angle \text{QOY}$
  • $\angle \text{QOY}$ and $\angle \text{POY}$
  • $\angle \text{POY}$ and $\angle \text{XOP}$
  • $\angle \text{XOP}$ and $\angle \text{XOQ}$

4. Vertically Opposite Angles

The two pairs of angles are formed by two intersecting lines are called vertically opposite angles. These pair of angles are opposite angles in such an intersection. Vertical opposite angles are always equal to each other.

linear pair of angles

In the above figure, the two intersecting lines $\text{AB}$ and $\text{XY}$, intersecting at $\text{O}$, forms two pairs of angles(or four angles) – $\angle \text{AOX}$, $\angle \text{XOB}$, $\angle \text{BOY}$, and $\angle \text{AOY}$.

In this case, $\angle \text{AOX}$ and $\angle \text{BOY}$ are vertically opposite angles and similarly, $\angle \text{AOY}$ and $\angle \text{XOB}$ are vertically opposite angles.

Therefore, $\angle \text{AOX} = \angle \text{BOY}$ and $\angle \text{AOY} = \angle \text{XOB}$.

Note: The angles in a pair of vertically opposite angles are always equal.

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5. Alternate Interior Angles

When a transversal line intersects a pair of parallel lines alternate interior angles are formed. Alternate interior angles are equal to each other. Alternate interior angles are formed on the inside of two parallel lines that are intersected by a transversal.

linear pair of angles

In the above figure, $\angle \text{3}$ and $\angle \text{5}$ form a pair of alternate interior angles. Similarly, $\angle \text{4}$ and $\angle \text{6}$ also forms a pair of alternate interior angles.

Note: 

  • When a transversal intersects a pair of parallel lines then alternate interior angles are equal.
  • When a transversal intersects a pair of non-parallel lines then alternate interior angles are not equal.

6. Alternate Exterior Angles

Alternate exterior angles are formed on either side of the transversal. One way to remember alternate exterior angles is that they are the vertical angles of the alternate interior angles.

linear pair of angles

In the above figure, $\angle \text{1}$ and $\angle \text{7}$ form a pair of alternate exterior angles. Similarly, $\angle \text{2}$ and $\angle \text{8}$ also forms a pair of alternate exterior angles.

Note: 

  • When a transversal intersects a pair of parallel lines then alternate exterior angles are supplementary.
  • When a transversal intersects a pair of non-parallel lines then alternate interior angles are not supplementary.

7. Corresponding Angles

When a pair of lines are intersected by a transversal line, then angles are formed on the same side of the transversal are called corresponding angles. One way to find the corresponding angles is to draw the letter $\text{F}$ on the diagram. The $\text{F}$ can also be facing backward.

linear pair of angles

In the above figure, the corresponding angles are

  • $\angle \text{1}$ and $\angle \text{5}$
  • $\angle \text{2}$ and $\angle \text{6}$
  • $\angle \text{3}$ and $\angle \text{7}$
  • $\angle \text{4}$ and $\angle \text{8}$

8. Adjacent Angles

When two angles are next to one another, they are called adjacent angles. Adjacent angles share a common side and a common vertex. Adjacent angles are often considered angle pairs, even though they have only one identifying property: they share a common vertex and side. They do not need to be complementary, supplementary, or special in any way.

linear pair of angles

In the above figure, $\angle \text{AOB}$ and $\angle \text{BOC}$ are adjacent angles as these angles share an arm (or side) $\text{OB}$.

Practice Problems

  1. What is meant by pair of angles?
  2. Define the following angles
    • Complementary Angles
    • Supplementary Angles
    • Linear Pair of Angles
    • Vertically Opposite Angles
    • Alternate Interior Angles
    • Alternate Exterior Angles
    • Corresponding Angles
    • Adjacent Angles
  3. State True or False
    • Angles in a pair of complementary angles are equal.
    • Sum of angles in a pair of complementary angles is $90^{\circ}$.
    • Sum of angles in a pair of complementary angles is $180^{\circ}$.
    • Angles in a pair of supplementary angles are equal.
    • Sum of angles in a pair of supplementary angles is $90^{\circ}$.
    • Sum of angles in a pair of supplementary angles is $180^{\circ}$.
    • Angles in a linear pair are equal.
    • Sum of angles in a linear pair is $90^{\circ}$.
    • Sum of angles in a linear pair is $180^{\circ}$.

FAQs

How many pairs of angles are there?

There are eight pairs of angles as listed below:
a) Complementary Angles
b) Supplementary Angles
c) Linear Pair of Angles
d) Vertically Opposite Angles
e) Alternate Interior Angles
f) Alternate Exterior Angles
g) Corresponding Angles
h) Adjacent Angles

What are complementary angles?

An angle pair that adds up to $90^{\circ}$ is called a pair of supplementary angles.

What are supplementary angles?

An angle pair that adds up to $180^{\circ}$ is called a pair of supplementary angles.

What are linear pairs of angles?

When two lines intersect each other, the adjacent angles make a linear pair. The sum of linear pairs is $180^{\circ}$. It should be noted that all linear pairs are supplementary because supplementary angles sum up to $180^{\circ}$.

Conclusion

The term ‘pair of angles’ is used for two angles taken together. Angle pairs are called that because they always appear as two angles working together to display some unusual or interesting property that helps in solving problems in Geometry.

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