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Supplementary Angles

Supplementary angles are two angles whose measures add up to 180°. In other words, if you have two angles, and when you add their measures together you get 180°, then those angles are considered supplementary. For example, if one angle measures 120°, the other angle must measure 60° to be supplementary to it.

Learn about, Supplementary Angles definition, examples and others in detail in this article.

Table of Content

What are Supplementary Angles?

Supplementary angles are two angles that, when added together, equal 180 degrees. For instance, if you have one angle measuring 60 degrees, its supplementary angle would measure 120 degrees, because 60 plus 120 equals 180 degrees.

The supplementary angles of 40 degrees and 140 degrees are shown in the image below:

supplementary-angles

Examples of Supplementary Angles

  • Two angles whose measures add up to 180°.

Straight angle with two right angles

  • Two angles with a shared vertex and a common side and non-common sides form a straight line.
  • Angles measuring 30° and 150° or 60° and 120° are among examples.
  • Geometric forms including straight lines, parallelograms, and triangles often have supplementary angles.

Properties of Supplementary Angles

Various properties of Supplemenytary Angles are:

  • Two supplementary angles are those that, when combined, have a sum equal to 180 degrees.
  • Some of them are located in place which are directly connected while others are located at places which are not directly joined.
  • What one angle is the sum of the other angle Given one angle and the supplement of that angle is another angle.
  • If two angles are equal to the sum of the same angle both are equal or congruent.
  • Two supplementary angles are those whose sum is equal to 180o and when laid side by side, their angle will span the entire line.
  • Anyone of you who is a right angle, then its supplement is also a right angle.
  • When discussing supplementary angle pairs, neither of the two angles can be acute and the other cannot be obtuse.

Adjacent and Non-adjacent Supplementary Angles

Differences between Adjacent and Non-adjacent Supplementary Angles are added in the table below:

Type

Definition

Example

Difference

Adjacent

Angles that share a common vertex and a common side, with non-common sides forming a straight line.

∠ABC and ∠CBD in a straight line

The angles are next to each other and share a side and vertex, forming a linear pair.

Non-adjacent

Angles whose measures sum up to 180° but do not share a common side.

∠PQR and ∠QRS in a parallelogram

The angles are not next to each other and do not share a side or vertex, but their sum equals 180 degrees.

How to Find Supplementary Angle?

To find supplementary angle of a given angle, subtract the measure of the given angle from 180°. For instance, if you have an angle measuring 60°, subtract 60 from 180:

180° – 60° = 120°

Thus, the supplementary angle for an angle measuring 60° is 120°. In general, if an angle measures x°, its supplementary angle would be 180° minus x°.

Theorem of Supplementary Angles

If two angles are supplementary to the same angle, then they are congruent.

Explanation: Let’s consider an angle ABC and two other angles, PQR and STU, which are both supplementary to angle ABC. This means that the sum of each angle with angle ABC equals 180 degrees.

  • Measure of angle ABC plus the measure of angle PQR equals 180°.
  • Measure of angle ABC plus the measure of angle STU equals 180°.

Since both equations involve the measure of angle ABC, we can deduce that the measure of angle PQR is equal to the measure of angle STU.

Therefore, if two angles are supplementary to the same angle, they are congruent to each other. This theorem is a fundamental concept in geometry and is often utilized in geometric proofs.

Complementary vs Supplementary Angles

Difference between complementary angles and supplementary angles is shown in table added below:

Aspect

Complementary Angles

Supplementary Angles

Definition

Two angles whose sum is 90° (a right-angle).

Two angles whose sum is 180° (a straight angle).

Sum

90°

180°

Example
  • 30° and 60°
  • 45° and 45°
  • 110° and 70°
  • 90° and 90°

Geometric Context

Often found in right triangles.

Often found in linear pairs or as adjacent angles forming a straight line.

Notation

 ∠A + ∠B = 90°

∠C + ∠D = 180°

Real-World Examples

Angles of a corner of a rectangular object.

Angles of a book opening flat.

Applications of Supplementary Angles

Real Life Applications of Supplementary Angles are:

  • Complementary and Supplementary Angles in Triangles.
  • Angles in Polygons.
  • Trigonometric Functions, etc.

Articles Related to Supplementary Angles:

Examples on Supplementary Angles

Example 1: Find the measure of two supplementary angles if one angle is 70 °.

Solution:

Two angles are supplementary if their sum is 180°.

Let the measure of the unknown angle be x.

Given: One angle = 70°

Since they are supplementary: 70 + x = 180

Subtract 70 from both sides: x = 180 – 70 x = 110

Thus, the measures of the two supplementary angles are 70° and 110°.

Example 2: If one of the supplementary angles is twice the other, find the measures of the angles.

Solution:

Let the measure of the smaller angle be x°.

Then, the measure of the larger angle is 2x°.

Since they are supplementary: x + 2x = 180

Combine like terms: 3x = 180

Divide both sides by 3: x = 60

So, the measure of the smaller angle is 60°, and the measure of the larger angle is: 2 x 60 = 120°

Thus, the measures of the two supplementary angles are 60° and 120°.

Example 3: The measure of one angle is 10 ° more than three times the measure of its supplement. Find the measures of both angles.

Solution:

Let the measure of one angle be x°.

Then, the measure of its supplement is 180 – x°.

Given: x = 3(180 – x) + 10

Expand and simplify: x = 540 – 3x + 10

Combine like terms: x + 3x = 550

4x = 550

Divide both sides by 4: x = 137.5

So, the measure of the first angle is 137.5°, and the measure of its supplement is: 180 – 137.5 = 42.5°

Thus, the measures of the two supplementary angles are 137.5° and 42.5°.

Example 4: The measure of one angle is 40° more than its supplement. Find the measures of the angles.

Solution:

Let the measure of the smaller angle be x°

Then, the measure of the larger angle is x + 40°

Since they are supplementary: x + (x + 40) = 180

Combine like terms: 2x + 40 = 180

Subtract 40 from both sides: 2x = 140

Divide both sides by 2: x = 70

So, the measure of the smaller angle is 70°, and the measure of the larger angle is: 70 + 40 = 110°

Thus, the measures of the two supplementary angles are 70° and 110°.

Example 5: If the difference between two supplementary angles is 30 °, find the measures of the angles.

Solution:

Let the measure of the larger angle be x°

Then, the measure of the smaller angle is x – 30°

Since they are supplementary: x + (x – 30) = 180

Combine like terms: 2x – 30 = 180

Add 30 to both sides: 2x = 210

Divide both sides by 2: x = 105

So, the measure of the larger angle is 105°, and the measure of the smaller angle is: 105 – 30 = 75°

Thus, the measures of the two supplementary angles are 105° and 75°.

Practice Questions on Supplementary Angles

Q1. If one angle measures 50°, what is the measure of its supplement?

Q2. If one angle is 3 times the measure of its supplement, find the measures of both angles.

Q3. The measure of one angle is 20° less than twice the measure of its supplement. Find the measures of both angles.

Q4. If the difference between two supplementary angles is 60°, find the measures of the angles.

Q5. One angle is 25° more than its supplement. Find the measures of both angles.

FAQs on Supplementary Angles

What are Supplementary Angles?

Supplementary angles are two angles whose sum is 180 degrees.

How to find Supplementary Angle?

To find a supplementary angle for a given angle, subtract its measure from 180 degrees.

Can Supplementary Angles be Adjacent?

Yes, adjacent supplementary angles share a common vertex and a common side.

Can Supplementary Angles be Non-Adjacent?

Yes, non-adjacent supplementary angles are located on different sides of a transversal line.

If Two Angles are Supplementary to Same Angle, what can be said about them?

They are congruent to each other.

Can both Supplementary Angles be Acute or Obtuse?

No, one angle must be acute while the other is obtuse.


 


Reffered: https://www.geeksforgeeks.org

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