Complementary and Supplementary Angles Lesson: Definition, Examples

Lesson Overview

• What Are Complementary Angles?
• What Are Supplementary Angles?
• How to Find If Two Angles Are Supplementary

Complementary Angles vs. Supplementary Angles

Understanding the differences between complementary and supplementary angles is key to mastering this topic.

• Complementary Angles: Sum up to 90° and typically form a right angle.
• Supplementary Angles: Sum up to 180° and form a straight line.

The main differences between the two are:

What Are Complementary Angles?

Complementary angles are two angles whose sum equals 90 degrees. These angles "complete" a right angle when combined.

Example: If you have an angle that measures 40°, the complementary angle must measure 50° because 40° + 50° = 90°. This sum of 90° is what defines complementary angles.

Imagine two pieces of paper that you place together at a right angle (90°). The two edges that meet at the corner represent the two angles. If one is 60°, the other must be 30° to form the right angle.

We have a point called O, and from this point, two lines extend upwards. One line goes towards point Q, and the other towards point P. There's also a straight line going from O to another point Q.

This creates two angles:

• The angle between points Q, O, and P is 50 degrees.
• The angle between points P, O, and Q (on the right side) is 40 degrees.

These two angles, QOP and POQ, add up to 90 degrees (50 + 40 = 90). When two angles add up to 90 degrees, we call them complementary angles.

Understanding Complementary Angles Using a Right Triangle

1. Right Triangle:
   • Picture a right triangle, which has one corner that is a perfect square (90 degrees):

Right Angle:

• The right angle (the square corner) is located at the bottom left corner of the triangle. It measures 90 degrees.

Right Angle = 90°

Remaining Angles:

• The other two corners (let's call them angle A and angle B) must be smaller to fit together and complete the triangle:

Angle A + Angle B = 90° (remaining degrees)

Adding the Angles:

• Since all three angles in a triangle add up to 180 degrees, and the right angle takes up 90 degrees, the other two angles share the remaining 90 degrees:

Angle A + Angle B = 90° (complementary angles)

How to Find If Two Angles Are Complementary

To find whether two angles are complementary, you simply add their measures. If the total is 90°, they are complementary.

• Example Problem: Let's check if 25° and 65° are complementary. Add them together: 25° + 65° = 90°. Since the sum is 90°, these two angles are complementary.
• Example with a Missing Angle: If you know one angle is 35°, what is the complementary angle? You subtract 35° from 90°, which gives you 55°. So, the complementary angle is 55°.

This simple process of addition and subtraction helps you quickly determine complementary angles in both mathematical problems and real-life scenarios.

What Are Supplementary Angles?

Supplementary angles are two angles whose sum is 180 degrees. These angles, when placed together, form a straight line, which makes them extremely useful in understanding linear shapes and structures.

Example: If one angle is 110°, the supplementary angle must be 70° because 110° + 70° = 180°. This total of 180° is the defining feature of supplementary angles.

Imagine a straight line on the ground. If you stand on that line and draw two angles from the same point, they must add up to 180° to form that straight line.

Imagine a perfectly straight line going from point Q on one side to point Q on the other. Right in the middle of that line is point O. Now, another line shoots upwards from point O to a point called P.

This makes two angles:

• A big angle between points Q, O, and P. This one is 140 degrees.
• A smaller angle between points P, O, and Q (on the other side). This one is 40 degrees.

Since the line QOQ is straight, the two angles (QOP and POQ) must add up to 180 degrees.

Understanding Supplementary Angles Using a Door

Closed Door:

• Picture a door standing straight up against the wall:

Opened Halfway:

• Now, imagine the door opened halfway:

Angles:

• When the door is open halfway, it forms two angles:
  • Angle A: Between the door and the wall at the hinges (90 degrees).
  • Angle B: Between the door and the wall at the doorknob (90 degrees).

Adding the Angles:

• Both angles can be expressed like this:

And as per the definition, when two angles add up to 180 degrees, they are called supplementary angles.

How to Find If Two Angles Are Supplementary

To determine if two angles are supplementary, simply add their measures. If their sum equals 180°, then they are supplementary angles.

• Example Problem: Check if 120° and 60° are supplementary. Add them together: 120° + 60° = 180°. Since the sum is 180°, these two angles are supplementary.
• Example with a Missing Angle: If one angle is 75°, the supplementary angle will be 180° - 75°, which equals 105°. So, the supplementary angle is 105°.

This method helps in identifying supplementary angles, whether you're solving a math problem or making measurements for real-life tasks like construction.

Take This Quiz :

Complementary and supplementary angles quiz