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°

Complementary Angles

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°

Supplementary Angles

∠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.


3. Adjacent Angles:

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.

Adjacent Angles

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.

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°.

Linear Pair

These are the pairs of angles in geometry.

Angle.

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.






5th Grade Geometry Page

5th Grade Math Problems

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