Supplementary Angles

Geometry is one of the important branches of mathematics that deals with the study of different shapes. It initiates the study of lines and angles. A straight line is a line without curves and it is defined as the shortest distance between two points. An angle is formed when the line segment meets at a point. There are different types of angles and the classification of pair of angles is given as:

  • Complementary angles
  • Supplementary angles
  • Corresponding Angles
  • Linear pairs
  • Alternate angles
  • Vertically opposite angles

Here, one of the pair of angles called supplementary angles explained in detail.

Supplementary Angles Definition

Two angles are said to be supplementary angles when they add up to 180 degrees. Two angles are supplementary, if

  • One of its angles is an acute angle and another angle is an obtuse angle.
  • Both of the angles are right angles.

This means that ∠A + ∠B = 180°.

See the figure below for a better understanding of the pair of angles which are supplementary.

Supplementary Angles

Difference between Complementary and Supplementary Angles

Complementary Angles Supplementary Angles
Sum of two angles is 90° Sum of two angles is 180°
Ex: ∠A + ∠B = 90°. Ex: ∠A + ∠B = 180°.
Complementary angles form a right-angled triangle. Supplementary angles form a straight line.

Properties of Supplementary Angles

The important properties of supplementary angles are:

  • The two angles are said to be supplementary angles when they add up to 180°.
  • The two angles together make a straight line, but the angles need not be together.
  • S” of supplementary angles stands for “Straight” line. This means they form 180°.

How to Find Supplementary Angles?

For example, if you had given that two angles form supplementary angles and you are provided with one angle and then asked to find the other angle, you can easily find the other angle using the formula.

∠A = 180° – ∠B (or)

∠B = 180° – ∠A

Adjacent and Non-Adjacent Supplementary Angles

Adjacent Supplementary Angles

The supplementary angles may be classified as either adjacent or non-adjacent. The adjacent supplementary angles share the common line segment or arm with each other, whereas the non-adjacent supplementary angles do not share the line segment or arm.

Supplementary Angles Theorem

The supplementary angle theorem states that if two angles are said to be supplementary to the same angle, then the two angles are said to be congruent.

Supplementary Angles Examples

Some of the examples of supplementary angles are:

  • 120° + 60° = 180°
  • 90° + 90° = 180°
  • 140° + 40° = 180°
  • 96° + 84° = 180°

Can 2 Acute Angles be Supplementary Angles?

No, two acute angles cannot form a supplementary angle.

By definition, acute angles are the angles that measure the angle greater than 0° and less than 90°. If you add two acute angles in which each angle is large as possible, its sum will be less than 180°. By the definition of supplementary angles, it is impossible to get the supplementary angle when we add two acute angles.

Example: 80° +60° = 140° which is not a supplementary angle.

But, in the case, if we add more than two acute angles, we can get supplementary angles.

Can 2 Obtuse Angles be Supplementary Angles?

No, two obtuse angles cannot form a supplementary angle.

By the definition of obtuse angles, the angles that measure greater than 90° are obtuse. If you add two obtuse angles, the sum will be greater than 180°. It will not satisfy the property of the supplementary angles when we add obtuse angles.

Example: 110° + 95° = 205° which is not a supplementary angle. [205° > 180° ]

Can 2 Right Angles be Supplementary Angles?

Yes, 2 right angles can form a supplementary angle. We know that when the measure of an angle is exactly 90°, then it is known as a right angle.

When two right angles are added, it is possible to get the supplementary angle. because 90° + 90° = 180°, as it satisfies the condition of supplementary angles.

Supplementary Angles problem

Question:

Find the measure of unknown angle from the given figure.

Supplementary Angle Example

Solution:

We know that the supplementary angles add up to 180°.

X + 55° + 40° = 180°

X + 95° = 180°

X = 180°- 95°

X = 85°

Therefore, the unknown angle, X = 85°

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