9.3 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

So far in this chapter, we have focused on solving word problems, which are similar to many real-world applications of algebra. In the next few sections, we will apply our problem-solving strategies to some common geometry problems.

Use the Properties of Angles

Are you familiar with the phrase ‘do a 180’? It means to 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 180 degrees. See Figure 9.5.

The image is a straight line with an arrow on each end. There is a dot in the center. There is an arrow pointing from one side of the dot to the other, and the angle is marked as 180 degrees.
Figure 9.5

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 9.6, ∠A is the angle with vertex at point A. The measure of ∠A is written mA.

The image is an angle made up of two rays. The angle is labeled with letter A.
Figure 9.6 ∠A is the angle with vertex at pointA.

We measure angles in degrees, and use the symbol ° to represent degrees. We use the abbreviation m for the measure of an angle. So if ∠A is 27°, we would write mA=27.

If the sum of the measures of two angles is 180°, then they are called supplementary angles. In Figure 9.7, each pair of angles is supplementary because their measures add to 180°. Each angle is the supplement of the other.

Part a shows a 120 degree angle next to a 60 degree angle. Together, the angles form a straight line. Below the image, it reads 120 degrees plus 60 degrees equals 180 degrees. Part b shows a 45 degree angle attached to a 135 degree angle. Together, the angles form a straight line. Below the image, it reads 45 degrees plus 135 degrees equals 180 degrees.
Figure 9.7 The sum of the measures of supplementary angles is 180°.

If the sum of the measures of two angles is 90°, then the angles are complementary angles. In Figure 9.8, each pair of angles is complementary, because their measures add to 90°. Each angle is the complement of the other.

Part a shows a 50 degree angle next to a 40 degree angle. Together, the angles form a right angle. Below the image, it reads 50 degrees plus 40 degrees equals 90 degrees. Part b shows a 60 degree angle attached to a 30 degree angle. Together, the angles form a right angle. Below the image, it reads 60 degrees plus 30 degrees equals 90 degrees.
Figure 9.8 The sum of the measures of complementary angles is 90°.

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.

Step 1. Read the problem and make sure you understand all the words and ideas. Draw a figure and label it with the given information.

Step 2. Identify what you are looking for.

Step 3. Name what you are looking for and choose a variable to represent it.

Step 4. Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.

Step 5. Solve the equation using good algebra techniques.

Step 6. Check the answer in the problem and make sure it makes sense.

Step 7. 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.

Did you notice that the words complementary and supplementary are in alphabetical order just like 90 and 180 are in numerical order?

Use the Properties of Triangles

What do you already know about triangles? Triangle have three sides and three angles. Triangles are named by their vertices. The triangle in Figure 9.9 is called ΔABC, read ‘triangle ABC’. We label each side with a lower case letter to match the upper case letter of the opposite vertex.

The vertices of the triangle on the left are labeled A, B, and C. The sides are labeled a, b, and c.

Figure 9.9 ΔABC has vertices A,B,andC and sides a,b,andc.

The three angles of a triangle are related in a special way. The sum of their measures is 180°.

Right Triangles

Some triangles have special names. We will look first at the right triangle. A right triangle has one 90° angle, which is often marked with the symbol shown in Figure 9.10.

A right triangle is shown. The right angle is marked with a box and labeled 90 degrees.
Figure 9.10

If we know that a triangle is a right triangle, we know that one angle measures 90° so we only need the measure of one of the other angles in order to determine the measure of the third angle.

In the examples so far, we could draw a figure and label it directly after reading the problem. In the next example, we will have to define one angle in terms of another. So we will wait to draw the figure until we write expressions for all the angles we are looking for.

Similar Triangles

When we use a map to plan a trip, a sketch to build a bookcase, or a pattern to sew a dress, we are working with similar figures. In geometry, if two figures have exactly the same shape but different sizes, we say they are similar figures. One is a scale model of the other. The corresponding sides of the two figures have the same ratio, and all their corresponding angles are have the same measures.

The two triangles in Figure 9.11 are similar. Each side of ΔABC is four times the length of the corresponding side of ΔXYZ and their corresponding angles have equal measures.

Two triangles are shown. They appear to be the same shape, but the triangle on the right is smaller. The vertices of the triangle on the left are labeled A, B, and C. The side across from A is labeled 16, the side across from B is labeled 20, and the side across from C is labeled 12. The vertices of the triangle on the right are labeled X, Y, and Z. The side across from X is labeled 4, the side across from Y is labeled 5, and the side across from Z is labeled 3. Beside the triangles, it says that the measure of angle A equals the measure of angle X, the measure of angle B equals the measure of angle Y, and the measure of angle C equals the measure of angle Z. Below this is the proportion 16 over 4 equals 20 over 5 equals 12 over 3.

Figure 9.11 ΔABC and ΔXYZ are similar triangles. Their corresponding sides have the same ratio and the corresponding angles have the same measure.

The length of a side of a triangle may be referred to by its endpoints, two vertices of the triangle. For example, in ΔABC:

We will often use this notation when we solve similar triangles because it will help us match up the corresponding side lengths.

Use the Pythagorean Theorem

The Pythagorean Theorem is a special property of right triangles that has been used since ancient times. It is named after the Greek philosopher and mathematician Pythagoras who lived around 500 BCE.

Remember that a right triangle has a 90° angle, which we usually mark with a small square in the corner. The side of the triangle opposite the 90° angle is called the hypotenuse, and the other two sides are called the legs. See Figure 9.12.

Three right triangles are shown. Each has a box representing the right angle. The first one has the right angle in the lower left corner, the next in the upper left corner, and the last one at the top. The two sides touching the right angle are labeled “leg” in each triangle. The sides across from the right angles are labeled “hypotenuse.”

Figure 9.12 In a right triangle, the side opposite the 90° angle is called the hypotenuse and each of the other sides is called a leg.

The Pythagorean Theorem tells how the lengths of the three sides of a right triangle relate to each other. It states that in any right triangle, the sum of the squares of the two legs equals the square of the hypotenuse.

Media

ACCESS ADDITIONAL ONLINE RESOURCES

  • Animation: The Sum of the Interior Angles of a Triangle
  • Similar Polygons
  • Example: Determine the Length of the Hypotenuse of a Right Triangle

Section 9.3 Exercises

Practice Makes Perfect

Use the Properties of Angles

In the following exercises, find ⓐ the supplement and ⓑ the complement of the given angle.

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

.

ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?