Supplementary and complementary angles

Supplementary angles

If the sum of linear angles at a common vertex is 180˚, then the angles are supplementary. Even if two right angles are added, equal to 180˚, the pair is known as a supplementary pair of angles.
If one angle out of a pair of linear angles is x, then the other angle is given by 180-x. This linearity remains the same for all the pairs of supplementary angles.

This property is also valid for trigonometric functions like-

Sin (180 – A) = Sin A
Cos (180 – A) = – Cos A (quadrant is changed)
Tan (180 – A) = – Tan A

From the given image, we can say that the pair of angles BOA and AOC are supplementary angles because their sum is equal to 180˚.
We can also find the other angle if only one of these angles is given. For example, consider only 60 degrees was given, we can find the other angle by 180-60 = 120˚.

Complementary angles

When the sum of two angles is equal to 90˚, then the angles are said to be complementary angles. So even if you add two angles and form a right angle, the two angles are known to be complementary.

Complementary angle phenomenon is valid in trigonometry as well. The ratios are given as-

sin (90°- A) = cos A and cos (90°- A) = sin A
tan (90°- A) = cot A and cot (90°- A) = tan A
sec (90°- A) = cosec A and cosec (90°- A) = sec A

In the above image, we can see that AOD and DOB are complementary angles because their sum is equal to 90˚.

Also, in this image, angles POQ and ABC can be called complementary angles because their sum adds up to 90 degrees.

Facts about complementary angles

Two right angles cannot complement each other. They will supplement each other.
Two obtuse angles cannot complement each other because their sum will be greater than 90 degrees.
Two complementary angles can be acute, but the reverse may not be valid.

Example

Find the values of angles P and Q, if angle P and angle Q are supplementary angles such that angle P = 2x+10 and angle Q is 6x-46

Solution

We know, the sum of angles of a supplementary pair is equal to 180˚.

Therefore, ∠P + ∠Q = 180˚

(2x + 10) + (6x - 46) = 180˚

8x - 36 =180

8x = 216

x = 27

Therefore, ∠P = 2(27) + 10 = 64˚ and ∠Q = 6(27) - 46 =116˚.

Example

Given that two angles are supplementary in nature. The value of the larger angle is 5 degrees more than 4 times the measure of the smaller angle. Find out the value of a larger angle in degrees.

Solution

We need to consider the two supplementary angles as x (larger) and y (smaller).

From the information above, x = 4y+5

We know the sum of angles is 180˚ if they are supplementary.

x + y = 180˚

(4y + 5) + y = 180˚

5y + 5 = 180˚

5y = 175˚ = 35˚

Therefore, the larger angle x = 4(35) + 5 = 145˚