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

Table of Contents
Use the Properties of Angles Supplementary and Complementary Angles How To Example 3.2.1 Your Turn 3.2.1 Example 3.2.2 Your Turn 3.2.2 Use the Properties of Triangles Sum of the Measures of the Angles of a Triangle Example 3.2.3 Your Turn 3.2.3 Example 3.2.4 Your Turn 3.2.4 Example 3.2.5 Your Turn 3.2.5 Properties of Similar Triangles Example 3.2.6 Your Turn 3.2.6 Use the Pythagorean Theorem The Pythagorean Theorem Example 3.2.7 Your Turn 3.2.7 Example 3.2.8 Your Turn 3.2.8 Example 3.2.9 Your Turn 3.2.9 Media
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    See Also
    Angles: online games, activities, and lessonsHow to Prove the Angle Sum Property of a Triangle: 7 StepsProperties of a Triangle - Formulas, Theorems, ExamplesThe Angle Sum Property in Geometry

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    Learning Objectives

    After completing this section, you should be able to:

    1. Use the properties of angles
    2. Use the properties of triangles
    3. Use the Pythagorean Theorem

    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’?

    3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (2)

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

    3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (3)

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

    If the sum of the measures of two angles is 180°, as in Figure 3.2.3, each pair of angles is supplementary because their measures add to 180°.180°. Each angle is the supplement of the other.

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

    If the sum of the measures of two angles is 90°, as in Figure 3.2.4, each pair of angles is complementary, because their measures add to 90°.90°. Each angle is the complement of the other.

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

    Supplementary and Complementary Angles

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

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

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

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

    In the next few sections, you will be introduced to some common geometry formulas. We will use the Problem Solving Strategy for Geometry Applications below. 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.

    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.

    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 3.2.1

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

    Answer
    ⓐ
    Step 1. Read the problem. Draw the figure and label it with the given information. 3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (6)
    Step 2. Identify what you are looking for. 3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (7)
    Step 3. Name. Choose a variable to represent it. 3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (8)
    Step 4. Translate.
    Write the appropriate formula for the situation and substitute in the given information.
    3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (9)
    3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (10)
    Step 5. Solve the equation. 3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (11)
    Step 6. Check. 3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (12)
    3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (13)
    Step 7. Answer the question. 3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (14)
    ⓑ
    Step 1. Read the problem. Draw the figure and label it with the given information. 3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (15)
    Step 2. Identify what you are looking for. 3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (16)
    Step 3. Name. Choose a variable to represent it. 3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (17)
    Step 4. Translate.
    Write the appropriate formula for the situation and substitute in the given information.
    3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (18)
    Step 5. Solve the equation. 3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (19)
    3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (20)
    Step 6. Check: 3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (21)
    3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (22)
    Step 7. Answer the question. 3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (23)

    Your Turn 3.2.1

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

    Did you notice that the words complementary and supplementary are in alphabetical order just like 9090 and 180180 are in numerical order?

    Example 3.2.2

    Two angles are supplementary. The larger angle is 30°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. 3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (24)
    Step 2. Identify what you are looking for. 3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (25)
    Step 3. Name. Choose a variable to represent it.
    The larger angle is 30° more than the smaller angle.    		 3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (26)
    3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (27)
    Step 4. Translate.
    Write the appropriate formula and substitute.
    3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (28)
    Step 5. Solve the equation. 3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (29)
    3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (30)
    3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (31)
    3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (32)
    3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (33)
    3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (34)
    3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (35)
    Step 6. Check: 3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (36)
    3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (37)
    3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (38)
    Step 7. Answer the question. 3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (39)

    Your Turn 3.2.2

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

    See Also
    8 Angles Games & Activities to Make Maths Class More Fun

    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 3.2.5 is called ΔABC,ΔABC, read ‘triangle ABCABC’. We label each side with a lower case letter to match the upper case letter of the opposite vertex.

    3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (40)

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

    m∠A+m∠B+m∠C=180°m∠A+m∠B+m∠C=180°

    Sum of the Measures of the Angles of a Triangle

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

    m∠A+m∠B+m∠C=180°m∠A+m∠B+m∠C=180°

    Example 3.2.3

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

    Answer
    Step 1. Read the problem. Draw the figure and label it with the given information. 3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (41)
    Step 2. Identify what you are looking for. 3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (42)
    Step 3. Name. Choose a variable to represent it. 3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (43)
    Step 4. Translate.
    Write the appropriate formula and substitute.
    3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (44)
    Step 5. Solve the equation. 3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (45)
    3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (46)
    3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (47)
    Step 6. Check: 3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (48)
    3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (49)
    Step 7. Answer the question. 3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (50)

    Your Turn 3.2.3

    The measures of two angles of a triangle are 31°31° and 128°.128°. 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°.

    3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (51)

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

    Example 3.2.4

    One angle of a right triangle measures 28°.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. 3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (52)
    Step 2. Identify what you are looking for. 3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (53)
    Step 3. Name. Choose a variable to represent it. 3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (54)
    Step 4. Translate.
    Write the appropriate formula and substitute.
    3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (55)
    Step 5. Solve the equation. 3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (56)
    3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (57)
    3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (58)
    Step 6. Check: 3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (59)
    3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (60)
    Step 7. Answer the question. 3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (61)

    Your Turn 3.2.4

    One angle of a right triangle measures 56°.56°. 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 3.2.5

    The measure of one angle of a right triangle is 20°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.

    		 3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (62)
    3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (63)
    3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (64)
    3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (65)
    Step 4. Translate.
    Write the appropriate formula and substitute into the formula.    		 3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (66)
    3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (67)
    Step 5. Solve the equation. 3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (68)
    3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (69)
    3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (70)
    3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (71)
    3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (72)
    3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (73)
    3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (74)
    Step 6. Check: 3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (75)
    3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (76)
    Step 7. Answer the question. 3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (77)

    Your Turn 3.2.5

    The measure of one angle of a right triangle is 50°50° 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 3.2.7 are similar. Each side of ΔABC is four times the length of the corresponding side of ΔXYZ and their corresponding angles have equal measures.

    3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (78)

    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.

    3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (79)

    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 3.2.6

    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.

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

    ABXY=BCYZ=ACXZ


    Since the side AB=4 corresponds to the side XY=3, we will use the ratio ABXY=43 to find the other sides.

    Be careful to match up corresponding sides correctly.
    3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (81)
    Step 5. Solve the equation. 3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (82)
    Step 6. Check: 3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (83)
    Step 7. Answer the question. The third side of ΔABC is 6 and the third side of ΔXYZ is 2.4.

    Your Turn 3.2.6

    ΔABC is similar to ΔXYZ. Find a.

    3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (84)

    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 500500 BCE. Remember that a right triangle has a 90° angle.

    3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (85)

    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 ΔABC,ΔABC,

    a2+b2=c2

    where cc is the length of the hypotenuse aa and bb are the lengths of the legs.

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

    To solve problems that use the Pythagorean Theorem, we will need to find square roots. As a review, 2525 is 55 because 52=25.52=25.

    Example 3.2.7

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

    3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (87)
    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.
    3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (88)
    Step 4. Translate.
    Write the appropriate formula.
    Substitute.
    3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (89)
    Step 5. Solve the equation. 3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (90)
    Step 6. Check: 3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (91)
    Step 7. Answer the question. The length of the hypotenuse is 5.

    Your Turn 3.2.7

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

    3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (92)

    Example 3.2.8

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

    3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (93)
    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
    3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (94)
    Step 4. Translate.
    Write the appropriate formula. Substitute.    		 3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (95)
    Step 5. Solve the equation. Isolate the variable term. Use the definition of the square root.
    Simplify.    		 3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (96)
    Step 6. Check: 3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (97)
    Step 7. Answer the question. The length of the leg is 12.

    Your Turn 3.2.8

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

    3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (98)

    Example 3.2.9

    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.

    3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (99)
    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
    3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (100)
    Step 4. Translate.
    Write the appropriate formula.
    Substitute.
    3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (101)
    Step 5. Solve the equation.
    Isolate the variable.
    Use the definition of the square root.
    Simplify. Approximate to the nearest tenth.    		 3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (102)
    Step 6. Check: 3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (103)
    Step 7. Answer the question. Kelvin should fasten each piece of wood approximately 7.1" from the corner.

    Your Turn 3.2.9

    John puts the base of a 13 foot-ft ladder 55 feet from the wall of his house. How far up the wall does the ladder reach?

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

    Media

    3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem (2024)
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    Introduction: My name is Virgilio Hermann JD, I am a fine, gifted, beautiful, encouraging, kind, talented, zealous person who loves writing and wants to share my knowledge and understanding with you.