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Mathematics LibreTexts

9.4: Use Properties of Angles, Triangles, and the Pythagorean Theorem (Part 1)

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    5007
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    Skills to Develop

    • Use the properties of angles
    • Use the properties of triangles
    • Use the Pythagorean Theorem

    be prepared!

    Before you get started, take this readiness quiz.

    1. Solve: x + 3 + 6 = 11. If you missed this problem, review Example 8.6.
    2. Solve: \(\frac{a}{45} = \frac{4}{3}\). If you missed this problem, review Example 6.42.
    3. 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’? 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 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 m ∠ A.

    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 point A.

    We measure angles in degrees, and use the symbol ° to represent degrees. We use the abbreviation m to 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 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°.

    Definition: 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 (a) its supplement, and (b) its complement.

    Solution

    (a)

    Step 1. Read the problem. Draw the figure and label it with the given information. CNX_BMath_Figure_09_03_047_img-01.png
    Step 2. Identify what you are looking for. the supplement of a 40°
    Step 3. Name. Choose a variable to represent it. let s = the measure of the supplement
    Step 4. Translate. Write the appropriate formula for the situation and substitute in the given information. $$m \angle A + m \angle B = 180$$
    Step 5. Solve the equation. $$\begin{split} s + 40 &= 180 \\ s &= 140 \end{split}$$
    Step 6. Check. $$\begin{split} 140 + 40 &\stackrel{?}{=} 180 \\ 180 &= 180\; \checkmark \end{split}$$
    Step 7. Answer the question. The supplement of the 40° angle is 140°.

    (b)

    Step 1. Read the problem. Draw the figure and label it with the given information. CNX_BMath_Figure_09_03_048_img-01.png
    Step 2. Identify what you are looking for. the complement of a 40°
    Step 3. Name. Choose a variable to represent it. let c = the measure of the complement
    Step 4. Translate. Write the appropriate formula for the situation and substitute in the given information. $$m \angle A + m \angle B = 90$$
    Step 5. Solve the equation. $$\begin{split} c + 40 &= 90 \\ c &= 50 \end{split}$$
    Step 6. Check. $$\begin{split} 50 + 40 &\stackrel{?}{=} 90 \\ 90 &= 90\; \checkmark \end{split}$$
    Step 7. Answer the question. The supplement of the 40° angle is 50°.

    Exercise 9.31:

    An angle measures 25°. Find (a) its supplement, and (b) its complement.

    Exercise 9.32:

    An angle measures 77°. Find (a) its supplement, and (b) its 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.

    Solution

    Step 1. Read the problem. Draw the figure and label it with the given information. CNX_BMath_Figure_09_03_049_img-01.png
    Step 2. Identify what you are looking for. the measures of both angles
    Step 3. Name. Choose a variable to represent it.

    let a = measure of the smaller angle

    a + 30 = measure of larger angle

    Step 4. Translate. Write the appropriate formula for the situation and substitute in the given information. $$m \angle A + m \angle B = 180$$
    Step 5. Solve the equation. $$\begin{split} (a + 30) + a &= 180 \\ 2a + 30 &= 180 \\ 2a &= 150 \\ a &= 75\quad measure\; of\; smaller\; angle \\ a &+ 30\quad measure\; of\; larger\; angle \\ 75 &+ 30 \\ &105 \end{split}$$
    Step 6. Check. $$\begin{split} m \angle A + m \angle B &= 180 \\ 75 + 105 &\stackrel{?}{=} 180 \\ 180 &= 180\; \checkmark \end{split}$$
    Step 7. Answer the question. The measures of the angles are 75° and 105°.

    Exercise 9.33:

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

    Exercise 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 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, 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°.

    $$m \angle A + m \angle B + m \angle C = 180°$$

    Definition: Sum of the Measures of the Angles of a Triangle

    For any ΔABC, the sum of the measures of the angles is 180°.

    $$m \angle A + m \angle B + m \angle C = 180°$$

    Example 9.18:

    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. CNX_BMath_Figure_09_03_050_img-01.png
    Step 2. Identify what you are looking for. the measure of the third angle in a triangle
    Step 3. Name. Choose a variable to represent it. let x = the measure of the angle
    Step 4. Translate. Write the appropriate formula for the situation and substitute in the given information. $$m \angle A + m \angle B + m \angle C = 180$$
    Step 5. Solve the equation. $$\begin{split} 55 + 82 + x &= 180 \\ 137 + x &= 180 \\ x &= 43 \end{split}$$
    Step 6. Check. $$\begin{split} 55 + 82 + 43 &\stackrel{?}{=} 180 \\ 180 &= 180\; \checkmark \end{split}$$
    Step 7. Answer the question. The measure of the third angle is 43 degrees.

    Exercise 9.35:

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

    Exercise 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° 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.

    Example 9.19:

    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. CNX_BMath_Figure_09_03_051_img-01.png
    Step 2. Identify what you are looking for. the measure of an angle
    Step 3. Name. Choose a variable to represent it. let x = the measure of the angle
    Step 4. Translate. Write the appropriate formula for the situation and substitute in the given information. $$m \angle A + m \angle B + m \angle C = 180$$
    Step 5. Solve the equation. $$\begin{split} x + 90 + 28 &= 180 \\ x + 118 &= 180 \\ x &= 62 \end{split}$$
    Step 6. Check. $$\begin{split} 180 &\stackrel{?}{=} 90 + 28 + 62 \\ 180 &= 180\; \checkmark \end{split}$$
    Step 7. Answer the question. The measure of the third angle is 62°.

    Exercise 9.37:

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

    Exercise 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.

    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.

    Let a = 1st angle

    a + 20 = 2nd angle

    90 = 3rd angle (the right angle)

    CNX_BMath_Figure_09_03_052_img-04.png

    Step 4. Translate. Write the appropriate formula for the situation and substitute in the given information. $$\begin{split} m \angle A + m \angle B + m \angle C &= 180 \\ a + (a + 20) + 90 &= 180 \end{split}$$
    Step 5. Solve the equation. $$\begin{split} 2a + 110 &= 180 \\ 2a &= 70 \\ a &= 35 \quad first\; angle \\ a + &20 \quad second\; angle \\ \textcolor{red}{35} + &20 \\ &55 \\ &90 \quad third\; angle \end{split}$$
    Step 6. Check. $$\begin{split} 35 + 55 + 90 &\stackrel{?}{=} 180 \\ 180 &= 180\; \checkmark \end{split}$$
    Step 7. Answer the question. The three angles measure 35°, 55°, and 90°.

    Exercise 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.

    Exercise 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 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.

    Definition: 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.

    ...

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

    the length a can also be written BC

    the length b can also be written AC

    the length c can also be written AB

    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:

    ΔABC and ΔXYZ are similar triangles. The lengths of two sides of each triangle are shown. Find the lengths of the third side of each triangle.

    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 a, the side across from B is labeled 3.2, and the side across from C is labeled 4. The vertices of the triangle on the right are labeled X, Y, and Z. The side across from X is labeled 4.5, the side across from Y is labeled y, and the side across from Z is labeled 3.

    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 ΔABC, y = length of the third side ΔXYZ
    Step 4. Translate.

    The triangles are similar, so the corresponding sides are in the same ratio. So$$\frac{AB}{XY} = \frac{BC}{YZ} = \frac{AC}{XZ}$$Since the side AB = 4 corresponds to the side XY = 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.

    CNX_BMath_Figure_09_03_057_img-01.png

    Step 5. Solve the equation. $$\begin{split} 3a &= 4(4.5) \qquad \; 4y = 3(3.2) \\ 3a &= 18 \qquad \qquad 4y = 9.6 \\ a &= 6 \qquad \qquad \quad y = 2.4 \end{split}$$
    Step 6. Check. $$\begin{split} \frac{4}{3} &\stackrel{?}{=} \frac{\textcolor{red}{6}}{4.5} \qquad \qquad \qquad \frac{4}{3} \stackrel{?}{=} \frac{3.2}{\textcolor{red}{2.4}} \\ 4(4.5) &\stackrel{?}{=} 6(3) \qquad \qquad \; 4(2.4) \stackrel{?}{=} 3.2(3) \\ 18 &= 18\; \checkmark \qquad \qquad \quad \; 9.6 = 9.6\; \checkmark \end{split}$$
    Step 7. Answer the question. The third side of ΔABC is 6 and the third side of ΔXYZ is 2.4.

    Exercise 9.41:

    ΔABC is similar to ΔXYZ. Find a.

    Two triangles are shown. They appear to be the same shape, but the triangle on the right is larger The vertices of the triangle on the left are labeled A, B, and C. The side across from A is labeled a, the side across from B is labeled 15, and the side across from C is labeled 17. The vertices of the triangle on the right are labeled X, Y, and Z. The side across from X is labeled 12, the side across from Y is labeled y, and the side across from Z is labeled 25.5.

    Exercise 9.42:

    ΔABC is similar to ΔXYZ. Find y

    Two triangles are shown. They appear to be the same shape, but the triangle on the right is larger The vertices of the triangle on the left are labeled A, B, and C. The side across from A is labeled a, the side across from B is labeled 15, and the side across from C is labeled 17. The vertices of the triangle on the right are labeled X, Y, and Z. The side across from X is labeled 12, the side across from Y is labeled y, and the side across from Z is labeled 25.5..

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