Complementary and Supplementary Angles
In this lesson, we will discuss the concepts of complementary and supplementary angles, key principles in geometry. We will provide clear definitions, six worked examples, exercises with solutions, a homework section, and a revision guide.
Outline
- Definitions of Complementary and Supplementary Angles
- Six examples with step-by-step solutions
- Exercises for practice
- Homework section
- Revision section
- Video explanation
1. Definitions
Complementary Angles
Complementary angles are two angles whose sum is \(90^\circ\). If two angles are complementary, they form a right angle.
Supplementary Angles
Supplementary angles are two angles whose sum is \(180^\circ\). These angles form a straight line when placed adjacent to each other.
2. Examples
Example 1: Identifying Complementary Angles
If one angle measures \(40^\circ\), what is the measure of its complementary angle?
Since the sum of complementary angles is \(90^\circ\):
\[ 90^\circ - 40^\circ = 50^\circ \]The complementary angle is \(50^\circ\).
Example 2: Identifying Supplementary Angles
If one angle measures \(110^\circ\), what is the measure of its supplementary angle?
Since the sum of supplementary angles is \(180^\circ\):
\[ 180^\circ - 110^\circ = 70^\circ \]The supplementary angle is \(70^\circ\).
Example 3: Complementary Angle in a Triangle
If two angles in a right triangle measure \(30^\circ\) and \(60^\circ\), are they complementary?
Yes, because their sum is:
\[ 30^\circ + 60^\circ = 90^\circ \]They form a right angle, so they are complementary.
Example 4: Supplementary Angles in a Linear Pair
If two angles form a straight line and one angle measures \(85^\circ\), what is the measure of the other angle?
The other angle is \(95^\circ\).
Example 5: Finding Missing Angles
If two angles are supplementary and one measures \(x^\circ\) while the other is \(2x^\circ\), find the value of \(x\).
Since the angles are supplementary, their sum is \(180^\circ\):
\[ x + 2x = 180^\circ \] \[ 3x = 180^\circ \] \[ x = 60^\circ \]The value of \(x\) is \(60^\circ\).
Example 6: Real-Life Application
If a clock shows the time 3:00 PM, what are the complementary and supplementary angles between the hour and minute hands?
The angle between the hour and minute hands at 3:00 PM is \(90^\circ\). So, the complementary angle is:
\[ 90^\circ - 90^\circ = 0^\circ \]There is no complementary angle since they already form a right angle. The supplementary angle is:
\[ 180^\circ - 90^\circ = 90^\circ \]3. Exercises
Exercise 1: If one angle measures \(75^\circ\), what is the measure of its complementary angle?
Exercise 2: If one angle measures \(125^\circ\), what is the measure of its supplementary angle?
Exercise 3: Two angles measure \(x^\circ\) and \(3x^\circ\) and are supplementary. Find \(x\).
Exercise 4: Find the measure of an angle that is complementary to an angle of \(42^\circ\).
Exercise 5: Find the measure of an angle that is supplementary to an angle of \(160^\circ\).
4. Homework
Complete the following problems for additional practice:
- Find the complementary angle for \(56^\circ\).
- Determine the supplementary angle for \(135^\circ\).
- Two angles are complementary and one is \(2x\) and the other is \(x\). Find \(x\).
- Calculate the supplementary angle of \(72^\circ\).
- In a right triangle, if one angle is \(45^\circ\), find the other two angles.
5. Revision
Remember that:
- Complementary angles add up to \(90^\circ\).
- Supplementary angles add up to \(180^\circ\).
- Complementary and supplementary angles can be used to solve various geometry problems.
6. Video Explanation
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