9.4: Use Properties of Angles, Triangles, and the Pythagorean Theorem (2024)

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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

Be Prepared 9.7

Before you get started, take this readiness quiz.

Solve: $x + 3 + 6 = 11 .$
If you missed this problem, review Example 8.6.

Be Prepared 9.8

Solve: $\frac{a}{45} = \frac{4}{3} .$
If you missed this problem, review Example 6.42.

Be Prepared 9.9

Simplify: $\sqrt{36 + 64} .$
If you missed this problem, review Example 5.72.

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 $180 ’? Figure 9.5.$

9.4: Use Properties of Angles, Triangles, and the Pythagorean Theorem (4)

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 $m ∠ A .$

9.4: Use Properties of Angles, Triangles, and the Pythagorean Theorem (5)

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 $m ∠ A = 27 .$

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

9.4: Use Properties of Angles, Triangles, and the Pythagorean Theorem (6)

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

9.4: Use Properties of Angles, Triangles, and the Pythagorean Theorem (7)

Supplementary and Complementary Angles

If the sum of the measures of two angles is $180°,$ then the angles are supplementary.

If $∠ A$ and $∠ B$ are supplementary, then $m ∠ A + m ∠ B = 180°.$

If the sum of the measures of two angles is $90°,$ then the angles are complementary.

If $∠ A$ and $∠ B$ are complementary, then $m ∠ A + m ∠ B = 90°.$

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.

Example 9.16

An angle measures $40° .$ Find ⓐ its supplement, and ⓑ its complement.

Answer

ⓐ
Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.
Step 3. Name. Choose a variable to represent it.
Step 4. Translate. Write the appropriate formula for the situation and substitute in the given information.
Step 5. Solve the equation.
Step 6. Check:
Step 7. Answer the question.

ⓑ
Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.
Step 3. Name. Choose a variable to represent it.
Step 4. Translate. Write the appropriate formula for the situation and substitute in the given information.
Step 5. Solve the equation.
Step 6. Check:
Step 7. Answer the question.

Try It 9.31

An angle measures $25° .$ Find its: ⓐ supplement ⓑ complement.

Try It 9.32

An angle measures $77° .$ Find its: ⓐ supplement ⓑ complement.

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

Example 9.17

Two angles are supplementary. The larger angle is $30°$ more than the smaller angle. Find the measure of both angles.

Answer

Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.
Step 3. Name. Choose a variable to represent it. The larger angle is 30° more than the smaller angle.
Step 4. Translate. Write the appropriate formula and substitute.
Step 5. Solve the equation.
Step 6. Check:
Step 7. Answer the question.

Try It 9.33

Two angles are supplementary. The larger angle is $100°$ more than the smaller angle. Find the measures of both angles.

Try It 9.34

Two angles are complementary. The larger angle is $40°$ more than the smaller angle. Find the measures of both angles.

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 $Δ A B C,$ read ‘triangle $ABC$ ’. We label each side with a lower case letter to match the upper case letter of the opposite vertex.

9.4: Use Properties of Angles, Triangles, and the Pythagorean Theorem (42)

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

$m ∠ A + m ∠ B + m ∠ C = 180°$

Sum of the Measures of the Angles of a Triangle

For any $Δ A B C,$ the sum of the measures of the angles is $180° .$

$m ∠ A + m ∠ B + m ∠ C = 180°$

Example 9.18

The measures of two angles of a triangle are $55°$ and $82° .$ Find the measure of the third angle.

Answer

Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.
Step 3. Name. Choose a variable to represent it.
Step 4. Translate. Write the appropriate formula and substitute.
Step 5. Solve the equation.
Step 6. Check:
Step 7. Answer the question.

Try It 9.35

The measures of two angles of a triangle are $31°$ and $128° .$ Find the measure of the third angle.

Try It 9.36

A triangle has angles of $49°$ and $75° .$ Find the measure of the third angle.

Right Triangles

Some triangles have special names. We will look first at the right triangle. A right triangle has one $90° Figure 9.10.$

9.4: Use Properties of Angles, Triangles, and the Pythagorean Theorem (53)

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.

Example 9.19

One angle of a right triangle measures $28° .$ What is the measure of the third angle?

Answer

Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.
Step 3. Name. Choose a variable to represent it.
Step 4. Translate. Write the appropriate formula and substitute.
Step 5. Solve the equation.
Step 6. Check:
Step 7. Answer the question.

Try It 9.37

One angle of a right triangle measures $56° .$ What is the measure of the other angle?

Try It 9.38

One angle of a right triangle measures $45° .$ What is the measure of the other 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.

Example 9.20

The measure of one angle of a right triangle is $20°$ more than the measure of the smallest angle. Find the measures of all three angles.

Answer

Step 1. Read the problem.
Step 2. Identify what you are looking for.	the measures of all three angles
Step 3. Name. Choose a variable to represent it. Now draw the figure and label it with the given information.
Step 4. Translate. Write the appropriate formula and substitute into the formula.
Step 5. Solve the equation.
Step 6. Check:
Step 7. Answer the question.

Try It 9.39

The measure of one angle of a right triangle is $50°$ more than the measure of the smallest angle. Find the measures of all three angles.

Try It 9.40

The measure of one angle of a right triangle is $30°$ more than the measure of the smallest angle. Find the measures of all three angles.

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 have the same measures.

The two triangles in Figure 9.11 are similar. Each side of $Δ A B C$ is four times the length of the corresponding side of $Δ X Y Z$ and their corresponding angles have equal measures.

9.4: Use Properties of Angles, Triangles, and the Pythagorean Theorem (80)

Properties of Similar Triangles

If two triangles are similar, then their corresponding angle measures are equal and their corresponding side lengths are in the same ratio.

9.4: Use Properties of Angles, Triangles, and the Pythagorean Theorem (81)

The length of a side of a triangle may be referred to by its endpoints, two vertices of the triangle. For example, in $Δ A B C :$

$\begin{matrix} the length a can also be written B C \\ the length b can also be written A C \\ the length c can also be written A B \end{matrix}$

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

Example 9.21

$Δ A B C$ and $Δ X Y Z$ are similar triangles. The lengths of two sides of each triangle are shown. Find the lengths of the third side of each triangle.

9.4: Use Properties of Angles, Triangles, and the Pythagorean Theorem (82)

Answer

Step 1. Read the problem. Draw the figure and label it with the given information.	The figure is provided.
Step 2. Identify what you are looking for.	The length of the sides of similar triangles
Step 3. Name. Choose a variable to represent it.	Let a = length of the third side of $Δ A B C$ y = length of the third side $Δ X Y Z$
Step 4. Translate.
The triangles are similar, so the corresponding sides are in the same ratio. So $\frac{A B}{X Y} = \frac{B C}{Y Z} = \frac{A C}{X Z}$ Since the side $A B = 4$ corresponds to the side $X Y = 3$ , we will use the ratio $\frac{AB}{XY} = \frac{4}{3}$ to find the other sides. Be careful to match up corresponding sides correctly.
Step 5. Solve the equation.
Step 6. Check:
Step 7. Answer the question.	The third side of $Δ A B C$ is 6 and the third side of $Δ X Y Z$ is 2.4.

Try It 9.41

$Δ A B C$ is similar to $Δ X Y Z .$ Find $a .$

9.4: Use Properties of Angles, Triangles, and the Pythagorean Theorem (86)

Try It 9.42

$Δ A B C$ is similar to $Δ X Y Z .$ Find $y .$

9.4: Use Properties of Angles, Triangles, and the Pythagorean Theorem (87)

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 $90° Figure 9.12.$

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.

The Pythagorean Theorem

In any right triangle $Δ A B C,$

$a^{2} + b^{2} = c^{2}$

where $c$ is the length of the hypotenuse $a$ and $b$ are the lengths of the legs.

9.4: Use Properties of Angles, Triangles, and the Pythagorean Theorem (89)

To solve problems that use the Pythagorean Theorem, we will need to find square roots. In Simplify and Use Square Roots we introduced the notation $\sqrt{m}$ and defined it in this way:

$If m = n^{2}, then \sqrt{m} = n for n \geq 0$

For example, we found that $\sqrt{25}$ is $5$ because $5^{2} = 25 .$

We will use this definition of square roots to solve for the length of a side in a right triangle.

Example 9.22

Use the Pythagorean Theorem to find the length of the hypotenuse.

9.4: Use Properties of Angles, Triangles, and the Pythagorean Theorem (90)

Answer

Step 1. Read the problem.
Step 2. Identify what you are looking for.	the length of the hypotenuse of the triangle
Step 3. Name. Choose a variable to represent it.	Let $c = the length of the hypotenuse$
Step 4. Translate. Write the appropriate formula. Substitute.
Step 5. Solve the equation.
Step 6. Check:
Step 7. Answer the question.	The length of the hypotenuse is 5.

Try It 9.43

Use the Pythagorean Theorem to find the length of the hypotenuse.

9.4: Use Properties of Angles, Triangles, and the Pythagorean Theorem (95)

Try It 9.44

Use the Pythagorean Theorem to find the length of the hypotenuse.

9.4: Use Properties of Angles, Triangles, and the Pythagorean Theorem (96)

Example 9.23

Use the Pythagorean Theorem to find the length of the longer leg.

9.4: Use Properties of Angles, Triangles, and the Pythagorean Theorem (97)

Answer

Step 1. Read the problem.
Step 2. Identify what you are looking for.	The length of the leg of the triangle
Step 3. Name. Choose a variable to represent it.	Let $b = the leg of the triangle$ Label side b
Step 4. Translate. Write the appropriate formula. Substitute.
Step 5. Solve the equation. Isolate the variable term. Use the definition of the square root. Simplify.
Step 6. Check:
Step 7. Answer the question.	The length of the leg is 12.

Try It 9.45

Use the Pythagorean Theorem to find the length of the leg.

9.4: Use Properties of Angles, Triangles, and the Pythagorean Theorem (102)

Try It 9.46

Use the Pythagorean Theorem to find the length of the leg.

9.4: Use Properties of Angles, Triangles, and the Pythagorean Theorem (103)

Example 9.24

Kelvin is building a gazebo and wants to brace each corner by placing a $10-inch$ wooden bracket diagonally as shown. How far below the corner should he fasten the bracket if he wants the distances from the corner to each end of the bracket to be equal? Approximate to the nearest tenth of an inch.

9.4: Use Properties of Angles, Triangles, and the Pythagorean Theorem (104)

Answer

Step 1. Read the problem.
Step 2. Identify what you are looking for.	the distance from the corner that the bracket should be attached
Step 3. Name. Choose a variable to represent it.	Let x = the distance from the corner
Step 4. Translate. Write the appropriate formula. Substitute.
Step 5. Solve the equation. Isolate the variable. Use the definition of the square root. Simplify. Approximate to the nearest tenth.
Step 6. Check: Yes.
Step 7. Answer the question.	Kelvin should fasten each piece of wood approximately 7.1" from the corner.

Try It 9.47

John puts the base of a $13-ft$ ladder $5$ feet from the wall of his house. How far up the wall does the ladder reach?

9.4: Use Properties of Angles, Triangles, and the Pythagorean Theorem (109)

Try It 9.48

Randy wants to attach a $17-ft$ string of lights to the top of the $15-ft$ mast of his sailboat. How far from the base of the mast should he attach the end of the light string?

9.4: Use Properties of Angles, Triangles, and the Pythagorean Theorem (110)

Media

ACCESS ADDITIONAL ONLINE RESOURCES

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.

81.

$53°$

Everyday Math

125.

Building a scale model Joe wants to build a doll house for his daughter. He wants the doll house to look just like his house. His house is $30$ feet wide and $35$ feet tall at the highest point of the roof. If the dollhouse will be $2.5$ feet wide, how tall will its highest point be?

126.

Measurement A city engineer plans to build a footbridge across a lake from point $X$ to point $Y,$ as shown in the picture below. To find the length of the footbridge, she draws a right triangle $XYZ,$ with right angle at $X .$ She measures the distance from $X$ to $Z, 800$ feet, and from $Y$ to $Z, 1,000$ feet. How long will the bridge be?

Writing Exercises

127.

Write three of the properties of triangles from this section and then explain each in your own words.

128.

Explain how the figure below illustrates the Pythagorean Theorem for a triangle with legs of length $3$ and $4 .$

Self Check

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

9.4: Use Properties of Angles, Triangles, and the Pythagorean Theorem (131)

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