Complementary Angles

When the sum of the measures of two angles is 90°, such angles are called complementary angles and each angle is called a complement of the other.

The vertices of two angles may be same or different. In the given figure ∠AOB and ∠BOC are complementary as ∠AOB + ∠BOC = 30° + 60° = 90°.

$omplementary angles$

Again, ∠PQR and ∠QRP are complementary as ∠PQR + ∠QRP = 40° + 50° = 90°.

$complementary angles$

Angles of measure 25° and 65° are complementary angles. The angle of 25° is the complement of the angle of 65° and the angle of 65° is the complement of the angle of 25°.

The complement of an angle of measure 32° is the angle of 58°. And, the complement of the angle of measure 58° is the angle of 32°.

Observations:

(i) If two are complement of each other, then each is an acute angle. But any two acute angles need not be complementary.

For example, angles of measure 30° and 50° are not complement of each other.

(ii) Two obtuse angles cannot be complement of each other.

(iii) Two right angles cannot be complement of each other.

Worked-out Problems on Complementary Angles:

1. Find the complement of:

(a) 68°

Solution:

90° - 68°

= 22°

Therefore, the complement of 68° is 22°

(b) 27°20'

Solution:

90° - 27°20'

= 89°60' - 27°20'

= 62°40'

Therefore, the complement of 27°20' is 62°40'

90° - (x + 52°)

= 90° - x + 52°

= 38° - x

Therefore, the complement of x + 52° is 38° - x

2. Find the complement of the angle (10 + y)°.

Solution:

Complement of the angle (10 + y)° = 90° - (10 + y)°

= 90° - 10° - y°

= (80 - y)°

3. Find the measure of an angle which is 46° less than its complement.

Solution:

Let the unknown angle be x, then measure of its complement = 90 - x

According to the question,

(90 - x) - x = 46°

90 - x - x = 46°

90 - 2x = 46°

90 - 90 - 2x = 46° - 90

-2x = 46° - 90

-2x = -44°

2x = 44°

x = 44/2

x = 22°

Therefore, 90 - x (Put the value of x = 22°)

= 90 - 22°