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°.
Again, ∠PQR and ∠QRP are complementary as ∠PQR + ∠QRP = 40° + 50° = 90°.
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'
(c) x + 52°
Solution:
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 = 46° - 90
-2x = -44°
2x = 44°
x = 44/2
x = 22°
Therefore, 90 - x (Put the value of x = 22°)
= 90 - 22°
= 68°
Therefore, the pair of complementary angles are 68° and 22°
● Lines and Angles
Fundamental Geometrical Concepts
Some Geometric Terms and Results
Complementary and Supplementary Angles
Parallel and Transversal Lines
7th Grade Math Problems
8th Grade Math Practice
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