CHAPTER 9 Trigonometry
9.1 Use Properties of Angles, Triangles, and the Pythagorean Theorem
Learning Objectives
By the end of this section, you will be able to:
- Use the properties of angles
- Use the properties of triangles
- Use the Pythagorean Theorem
Use the Properties of Angles
Are you familiar with the phrase ‘do a It means to make a full turn so that you face the opposite direction. It comes from the fact that the measure of an angle that makes a straight line is degrees. See (Figure 1).
An angle is formed by two rays that share a common endpoint. Each ray is called a side of the angle and the common endpoint is called the vertex. An angle is named by its vertex. In (Figure 2), is the angle with vertex at point . The measure of is written .
We measure angles in degrees, and use the symbol ° to represent degrees. We use the abbreviation to for the measure of an angle. So if is 27°, we would write .
If the sum of the measures of two angles is °, then they are called supplementary angles. In (Figure 3), each pair of angles is supplementary because their measures add to °. Each angle is the supplement of the other.
If the sum of the measures of two angles is °, then the angles are complementary angles. In (Figure 4), each pair of angles is complementary, because their measures add to °. Each angle is the complement of the other.
Supplementary and Complementary Angles
If the sum of the measures of two angles is °, then the angles are supplementary.
If and are supplementary, then °.
If the sum of the measures of two angles is °, then the angles are complementary.
If and are complementary, then °.
In this section and the next, you will be introduced to some common geometry formulas. We will adapt our Problem Solving Strategy for Geometry Applications. The geometry formula will name the variables and give us the equation to solve.
In addition, since these applications will all involve geometric shapes, it will be helpful to draw a figure and then label it with the information from the problem. We will include this step in the Problem Solving Strategy for Geometry Applications.
HOW TO: Use a Problem Solving Strategy for Geometry Applications
- Read the problem and make sure you understand all the words and ideas. Draw a figure and label it with the given information.
- Identify what you are looking for.
- Name what you are looking for and choose a variable to represent it.
- Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.
- Solve the equation using good algebra techniques.
- Check the answer in the problem and make sure it makes sense.
- Answer the question with a complete sentence.
The next example will show how you can use the Problem Solving Strategy for Geometry Applications to answer questions about supplementary and complementary angles.
EXAMPLE 1
An angle measures °. Find a) its supplement, and b) its complement.
| a) | |
| Step 1. Read the problem. Draw the figu