Use Properties of Angles, Triangles, and 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 1.

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.

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, $∠ A$ is the angle with vertex at point $A .$ The measure of $∠ A$ is written $m ∠ A .$

The image is an angle made up of two rays. The angle is labeled with letter A. — $∠ A$ is the angle with vertex at $point A .$

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 $180°,$ then they are called supplementary angles. In Figure 3, 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. — 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 4, 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. — 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.

The next example will show how you can use the Problem Solving Strategy for Geometry Applications to answer questions about supplementary and complementary angles.

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

Solution

ⓐ
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.

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

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

Solution

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.

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 5 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.

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

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°

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

Solution

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.

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 6.

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

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.

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

Solution

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.

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.

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.

Solution

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.

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 7 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.

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. — $Δ A B C$ and $Δ X Y Z$ 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 $Δ A B C :$

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

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

$Δ A B C$ 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.

Solution

Step-by-step guide demonstrating the process of finding unknown side lengths in similar triangles, from problem interpretation to solution.
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: $Two examples demonstrate how to check if two fractions are proportional by using the cross-multiplication method, showing the resulting equalities with a checkmark.$
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.

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 8.

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.” — 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.

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.

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

Right triangle with legs labeled as 3 and 4.

Solution

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.

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

Right triangle is shown with one leg labeled as 5 and hypotenuse labeled as 13.

Solution

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.

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.

A picture of a gazebo is shown. Beneath the roof is a rectangular shape. There are two braces from the top to each side. The brace on the left is labeled as 10 inches. From where the brace hits the side to the roof is labeled as x.

Solution

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.

Key Concepts

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 $m ∠ A + m ∠ B = 90$ .
Solve Geometry Applications
1. Read the problem and make sure you understand all the words and ideas. Draw a figure and label it with the given information.
2. Identify what you are looking for.
3. Name what you are looking for and choose a variable to represent it.
4. Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.
5. Solve the equation using good algebra techniques.
6. Check the answer in the problem and make sure it makes sense.
7. Answer the question with a complete sentence.
Sum of the Measures of the Angles of a Triangle
- For any $Δ A B C,$ the sum of the measures is 180°
- $m ∠ A + m ∠ B + m ∠ C = 180$
Right Triangle
- A right triangle is a triangle that has one 90° angle, which is often marked with a $⦜$ symbol.
Properties of Similar Triangles
- If two triangles are similar, then their corresponding angle measures are equal and their corresponding side lengths have the same ratio.

Practice Makes Perfect

Use the Properties of Angles

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

$53°$

Solution

ⓐ 127°
ⓑ 37°

$16°$

$29°$

Solution

ⓐ 151°
ⓑ 61°

$72°$

In the following exercises, use the properties of angles to solve.

Find the supplement of a $135°$ angle.

Solution

45°

Find the complement of a $38°$ angle.

Find the complement of a $27.5°$ angle.

Solution

62.5°

Find the supplement of a $109.5°$ angle.

Two angles are supplementary. The larger angle is $56°$ more than the smaller angle. Find the measures of both angles.

Solution

62°, 118°

Two angles are supplementary. The smaller angle is $36°$ less than the larger angle. Find the measures of both angles.

Two angles are complementary. The smaller angle is $34°$ less than the larger angle. Find the measures of both angles.

Solution

62°, 28°

Two angles are complementary. The larger angle is $52°$ more than the smaller angle. Find the measures of both angles.

Use the Properties of Triangles

In the following exercises, solve using properties of triangles.

The measures of two angles of a triangle are $26°$ and $98° .$ Find the measure of the third angle.

Solution

56°

The measures of two angles of a triangle are $61°$ and $84° .$ Find the measure of the third angle.

The measures of two angles of a triangle are $105°$ and $31° .$ Find the measure of the third angle.

Solution

44°

The measures of two angles of a triangle are $47°$ and $72° .$ Find the measure of the third angle.

One angle of a right triangle measures $33° .$ What is the measure of the other angle?

Solution

57°

One angle of a right triangle measures $51° .$ What is the measure of the other angle?

One angle of a right triangle measures $22.5 ° .$ What is the measure of the other angle?

Solution

67.5°

One angle of a right triangle measures $36.5 ° .$ What is the measure of the other angle?

The two smaller angles of a right triangle have equal measures. Find the measures of all three angles.

Solution

45°, 45°, 90°

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

The angles in a triangle are such that the measure of one angle is twice the measure of the smallest angle, while the measure of the third angle is three times the measure of the smallest angle. Find the measures of all three angles.

Solution

30°, 60°, 90°

The angles in a triangle are such that the measure of one angle is $20°$ more than the measure of the smallest angle, while the measure of the third angle is three times the measure of the smallest angle. Find the measures of all three angles.

Find the Length of the Missing Side

In the following exercises, $Δ A B C$ is similar to $Δ X Y Z .$ Find the length of the indicated side.

side $b$

Solution

side $x$

On a map, San Francisco, Las Vegas, and Los Angeles form a triangle whose sides are shown in the figure below. The actual distance from Los Angeles to Las Vegas is $270$ miles.
A triangle is shown. The vertices are labeled San Francisco, Las Vegas, and Los Angeles. The side across from San Francisco is labeled 1 inch, the side across from Las Vegas is labeled 1.3 inches, and the side across from Los Angeles is labeled 2.1 inches.

Find the distance from Los Angeles to San Francisco.

Solution

351 miles

Find the distance from San Francisco to Las Vegas.

Use the Pythagorean Theorem

In the following exercises, use the Pythagorean Theorem to find the length of the hypotenuse.

A right triangle is shown. The right angle is marked with a box. One of the sides touching the right angle is labeled as 9, the other as 12.

Solution

A right triangle is shown. The right angle is marked with a box. One of the sides touching the right angle is labeled as 16, the other as 12.

A right triangle is shown. The right angle is marked with a box. One of the sides touching the right angle is labeled as 15, the other as 20.

Solution

A right triangle is shown. The right angle is marked with a box. One of the sides touching the right angle is labeled as 5, the other as 12.

Find the Length of the Missing Side

In the following exercises, use the Pythagorean Theorem to find the length of the missing side. Round to the nearest tenth, if necessary.

A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 10. One of the sides touching the right angle is labeled as 6.

Solution

A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 17. One of the sides touching the right angle is labeled as 8.

A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 13. One of the sides touching the right angle is labeled as 5.

Solution

A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 20. One of the sides touching the right angle is labeled as 16.

Solution

10.2

A right triangle is shown. The right angle is marked with a box. Both of the sides touching the right angle are labeled as 6.

Solution

16.2

A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 7. One of the sides touching the right angle is labeled as 5.

In the following exercises, solve. Approximate to the nearest tenth, if necessary.

A $13-foot$ string of lights will be attached to the top of a $12-foot$ pole for a holiday display. How far from the base of the pole should the end of the string of lights be anchored?

A vertical pole is shown with a string of lights going from the top of the pole to the ground. The pole is labeled 12 feet. The string of lights is labeled 13 feet.

Solution

5 feet

Pam wants to put a banner across her garage door to congratulate her son on his college graduation. The garage door is $12$ feet high and $16$ feet wide. How long should the banner be to fit the garage door?

A picture of a house is shown. The rectangular garage is 12 feet high and 16 feet wide. A blue banner goes diagonally across the garage.

Chi is planning to put a path of paving stones through her flower garden. The flower garden is a square with sides of $10$ feet. What will the length of the path be?

A square garden is shown. One side is labeled as 10 feet. There is a diagonal path of blue circular stones going from the lower left corner to the upper right corner.

Solution

14.1 feet

Brian borrowed a $20-foot$ extension ladder to paint his house. If he sets the base of the ladder $6$ feet from the house, how far up will the top of the ladder reach?

A picture of a house is shown with a ladder leaning against it. The ladder is labeled 20 feet tall. The horizontal distance from the house to the base of the ladder is 6 feet.

Everyday Math

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?

Solution

2.9 feet

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?

A lake is shown. Point Y is on one side of the lake, directly across from point X. Point Z is on the same side of the lake as point X.

Writing Exercises

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

Solution

Answers will vary.

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

Three squares are shown, forming a right triangle in the center. Each square is divided into smaller squares. The smallest square is divided into 9 small squares. The medium square is divided into 16 small squares. The large square is divided into 25 small squares.

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?

angle

An angle is formed by two rays that share a common endpoint. Each ray is called a side of the angle.

complementary angles

If the sum of the measures of two angles is

90°

, then they are called complementary angles.

hypotenuse

The side of the triangle opposite the 90° angle is called the hypotenuse.

legs of a right triangle

The sides of a right triangle adjacent to the right angle are called the legs.

right triangle

A right triangle is a triangle that has one

90°

angle.

similar figures

In geometry, if two figures have exactly the same shape but different sizes, we say they are similar figures.

supplementary angles

If the sum of the measures of two angles is

180°

, then they are called supplementary angles.

triangle

A triangle is a geometric figure with three sides and three angles.

vertex of an angle

When two rays meet to form an angle, the common endpoint is called the vertex of the angle.