Measure of supplementary angle of 5 3 part of two right angle.

Supplementary angles are pairs angles such that the sum of their angles is equal to 180 degrees.

Although the angle measurement of straight is equal to 180 degrees, a straight angle can’t be called a supplementary angle because the angle only appears in a single form. For angles to be called supplementary, they must add up to 180° and appear in pairs.

Possibilities of a supplementary angle

An acute and obtuse angle

A supplementary angle can be composed of one acute angle and another obtuse angle.

Illustration:

∠ θ and ∠ β are supplementary angles because they add up to 180 degrees. ∠ θ is an acute angle, while ∠ β is an obtuse angle.

∠ θ and ∠ β are also adjacent angles because they share a common vertex and arm.

An acute angle is an angle whose measure of degree is more than zero degrees but less than 90 degrees.

On the other hand, an obtuse angle is an angle whose measure of degree is more than 90 degrees but less than 180 degrees.

Common examples of supplementary angles of this type include:

⟹ 120° and 60°

⟹ 30° and 150°

⟹ 100° + 80°

⟹ 140° and 40°

⟹ 160° and 20° etc.

A supplementary angle can be made up of two right angles. A right angle is an angle that is exactly 90 degrees.

Illustration:

Non-adjacent supplementary angles

Two pairs of supplementary angles don’t have to be in the same figure.

Illustration:

The two angles in the above separate figures are complementary, i.e., 1400 + 400 = 1800

How to Find Supplementary Angles?

We can calculate supplementary angles by subtracting the given one angle from 180 degrees. To find the other angle, use the following formula:

∠x = 180° – ∠y or ∠y = 180° – ∠x where ∠x or ∠y is the given angle.

Let’s work on the following examples.

Example 1

Check whether the angles 127° and 53° are a pair of supplementary angles.

Solution

127° + 53° = 180°

Hence, 127° and 53° are pairs of supplementary angles.

Example 2

Check if the two angles, 170°, and 19° are supplementary angles.

Solution

170° + 19° = 189°

Since 189°≠ 180°, therefore, 170° and 19° are not supplementary angles.

Example 3

Given two supplementary angles as: (x – 2) ° and (x + 5) °, determine the value of x.

Solution

The sum of the angles must be equal to 180 degrees: (x – 2) + (2x + 5) = 180

⟹ x – 2 + 2x + 5 = 180

⟹ x + 2x – 2 + 5 = 180

⟹ 3x + 3 = 180

⟹ 3x + 3 – 3 = 180 — 3

⟹ 3x = 180 — 3

⟹ 2x = 177

Divide both sides by 3 to get x as;

x = 59°
Therefore, the value of x is 59°.

Example 4

Calculate the value of θ in the figure below.

Solution

⟹ (5θ + 4°) + (θ – 2°) + (3θ + 7°) = 180°

⟹ 5θ + 4° + θ – 2° + 3θ + 7° = 180°

⟹ 5θ + θ + 3θ + 4° – 2° + 7° = 180°

⟹ 9θ + 9° = 180°

⟹ 9θ + 9° – 9° = 180° – 9°

⟹ 9θ = 171°

⟹ θ = 171/9

⟹ θ = 19°

Example 5

The ratio of a pair of supplementary angles is 1:8. Find the two measures of the two angles?

Solution

Let r be the common ratio.

One angle will be r, and the other will be 8r

Therefore, r + 8r = 180.

9r = 180

r = 180/9

r = 20

Substitute r = 20 in the initial equations.

Hence, one angle is 20 degrees, and the other is 160 degrees.

Therefore, the angles 20 degrees and 160 degrees are the two supplementary angles.

Example 6

Determine the supplement angle of (x + 10) °.

Solution

⟹ (x + 10) ° = 180 ° – (x + 10) °

= 180° – 10° – x°

= (170 – x) °

Before we solve the worked-out problems on complementary and supplementary angles we will recall the definition of complementary angles and supplementary angles.

Complementary Angles:

Two angles are called complementary angles, if their sum is one right angle i.e. 90°.

Each angle is called the complement of the other. Example, 20° and 70° are complementary angles, because 20° + 70° = 90°.

Clearly, 20° is the complement of 70° and 70° is the complement of 20°.Thus, the complement of angle 53° = 90° - 53° = 37°.

Supplementary Angles:

Two angles are called supplementary angles, if their sum is two right angles i.e. 180°.

Each angle is called the supplement of the other. Example, 30° and 150° are supplementary angles, because 30° + 150° = 180°.

Clearly, 30° is the supplement of 150° and 150° is the supplement of 30°.

Thus, the supplement of angle 105° = 180° - 105° = 75°.

Solved problems on complementary and supplementary angles:

1. Find the complement of the angle 2/3 of 90°.

Solution:

Convert 2/3 of 90°

2/3 × 90° = 60°

Complement of 60° = 90° - 60° = 30°

Therefore, complement of the angle 2/3 of 90° = 30°

2. Find the supplement of the angle 4/5 of 90°.

Solution:

Convert 4/5 of 90°

4/5 × 90° = 72°

Supplement of 72° = 180° - 72° = 108°

Therefore, supplement of the angle 4/5 of 90° = 108°

3. The measure of two complementary angles are (2x - 7)° and (x + 4)°. Find the value of x.

Solution:

According to the problem, (2x - 7)° and (x + 4)°, are complementary angles’ so we get;

(2x - 7)° + (x + 4)° = 90°

or, 2x - 7° + x + 4° = 90°

or, 2x + x - 7° + 4° = 90°

or, 3x - 3° = 90°

or, 3x - 3° + 3° = 90° + 3°

or, 3x = 93°

or, x = 93°/3°

or, x = 31°

Therefore, the value of x = 31°.

4. The measure of two supplementary angles are (3x + 15)° and (2x + 5)°. Find the value of x.

Solution:

According to the problem, (3x + 15)° and (2x + 5)°, are complementary angles’ so we get;

(3x + 15)° + (2x + 5)° = 180°

or, 3x + 15° + 2x + 5° = 180°

or, 3x + 2x + 15° + 5° = 180°

or, 5x + 20° = 180°

or, 5x + 20° - 20° = 180° - 20°

or, 5x = 160°

or, x = 160°/5°

or, x = 32°

Therefore, the value of x = 32°.

5. The difference between the two complementary angles is 180°. Find the measure of the angle.

Solution:

Let one angle be of measure x°.

Then complement of x° = (90 - x)

Difference = 18°

Therefore, (90° - x) – x = 18°

or, 90° - 2x = 18°

or, 90° - 90° - 2x = 18° - 90°

or, -2x = -72°

or, x = 72°/2°

or, x = 36°

Also, 90° - x

= 90° - 36°

= 54°.

Therefore, the two angles are 36°, 54°.

6. POQ is a straight line and OS stands on PQ. Find the value of x and the measure of ∠ POS, ∠ SOR and ∠ ROQ.

Solution:

POQ is a straight line.

Therefore, ∠POS + ∠SOR + ∠ROQ = 180°

or, (5x + 4°) + (x - 2°) + (3x + 7°) = 180°

or, 5x + 4° + x - 2° + 3x + 7° = 180°

or, 5x + x + 3x + 4° - 2° + 7° = 180°

or, 9x + 9° = 180°

or, 9x + 9° - 9° = 180° - 9°

or, 9x = 171°

or, x = 171/9

or, x = 19° Put the value of x = 19°

Therefore, x - 2

= 19 - 2

= 17° Again, 3x + 7

= 3 × 19° + 7°

= 570 + 7°

= 64° And again, 5x + 4

= 5 × 19° + 4°

= 95° + 4°

= 99°

Therefore, the measure of the three angles is 17°, 64°, 99°. These are the above solved examples on complementary and supplementary angles explained step-by-step with detailed explanation.

● Lines and Angles

Fundamental Geometrical Concepts

Angles

Classification of Angles

Related Angles

Some Geometric Terms and Results

Complementary Angles

Supplementary Angles

Complementary and Supplementary Angles

Adjacent Angles

Linear Pair of Angles

Vertically Opposite Angles