# Pairs of Angles

Pairs of angles are discussed here in this lesson.

1. Complementary Angles:

Two angles whose sum is 90° (that is, one right angle) are called complementary angles and one is called the complement of the other.

Here, ∠AOB = 40° and ∠BOC = 50°

Therefore, ∠AOB + ∠BOC = 90°

Here, ∠AOB and ∠BOC are called complementary angles.

∠AOB is complement of ∠BOC and ∠BOC is complement of ∠AOB.

For Example:

(i) Angles of measure 60° and 30° are complementary angles because 60° + 30° = 90°

Thus, the complementary angle of 60° is the angle measure 30°. The complementary angle angle of 30° is the angle of measure 60°.

(ii) Complement of 30° is → 90° - 30° = 60°

(iii) Complement of 45° is → 90° - 45° = 45°

(iv) Complement of 55° is → 90° - 55° = 35°

(v) Complement of 75° is → 90° - 75° = 15°

Working rule: To find the complementary angle of a given angle subtract the measure of an angle from 90°.

So, the complementary angle = 90° - the given angle.

2. Supplementary Angles:

Two angles whose sum is 180° (that is, one straight angle) are called supplementary angles and one is called the supplement of the other.

Here, ∠PQR = 50° and ∠RQS = 130°

∠PQR + ∠RQS = 180° Hence, ∠PQR and ∠RQS are called supplementary angles and ∠PQR is

supplement of ∠RQS and ∠RQS is supplement of ∠PQR.

For Example:

(i) Angles of measure 100° and 80° are supplementary angles because 100° + 80° = 180°.

Thus the supplementary angle of 80° is the angle of measure 100°.

(ii) Supplement of - 55° is 180° - 55° = 125°

(iii) Supplement of 95° is 180° - 95° = 85°

(iv) Supplement of 135° is 180° - 135° = 45°

(v) Supplement of 150° is 180° - 150° = 30°

Working rule: To find the supplementary angle of a given angle, subtract the measure of angle from 180°.

So, the supplementary angle = 180° - the given angle.

Two non – overlapping angles are said to be adjacent angles if they have:

(a) a common vertex

(b) a common arm

(c) other two arms lying on opposite side of this common arm, so that their interiors do not overlap.

In the above given figure, ∠AOB and ∠BOC are non – overlapping, have OB as the common arm and O as the common vertex. The other arms OC and OA of the angles ∠BOC and ∠AOB are an opposite sides, of the common arm OB.

Hence, the arm ∠AOB and ∠BOC form a pair of adjacent angles.

4. Vertically Opposite Angles:

Two angles formed by two intersecting lines having no common arm are called vertically opposite angles.

In the above given figure, two lines $$\overleftrightarrow{AB}$$ and $$\overleftrightarrow{CD}$$ intersect each other at a point O.

They form four angles ∠AOC, ∠COB, ∠BOD and ∠AOD in which ∠AOC and ∠BOD are vertically opposite angles. ∠COB and ∠AOD are vertically opposite angle.

∠AOC and ∠COB, ∠COB and ∠BOD, ∠BOD and ∠DOA, ∠DOA and ∠AOC are pairs of adjacent angles.

Similarly we can say that, ∠1 and ∠2 form a pair of vertically opposite angles while ∠3 and ∠4 form another pair of vertically opposite angles.

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

∠1 = ∠2

∠3 = ∠4

5. Linear Pair:

Two adjacent angles are said to form a linear pair if their sum is 180°.

These are the pairs of angles in geometry.

Interior and Exterior of an Angle.

Measuring an Angle by a Protractor.

Types of Angles.

Pairs of Angles.

Bisecting an angle.

Construction of Angles by using Compass.

Worksheet on Angles.

Geometry Practice Test on angles.