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Transcript
Lesson 21 Understand Angle Relationships Name: Prerequisite: How can you use facts about angles to help you find a missing angle measure? Study the example showing how to find the measure of a missing angle. Then solve problems 1–6. Example Triangle ABC is a right triangle whose right angle is ABC. Find the measure of EBF. ABC and DBF are vertical angles, so they have the same measure. Because m ABC is 90°, the sum of m DBE and m EBF must also be 90°. Solve for x in this equation. D (x 2 12)8 E A B x8 F G x 1 (x 2 12) 5 90 2x 2 12 5 90 C 2x 5 102 x 5 51 m EBF 5 51° 1 What is m DBE? Explain. 2 What is m GBC? Explain. 3 Explain how to find m ABD. Vocabulary complementary angles two angles whose measures add to 90°. supplementary angles two angles whose measures add to 180°. vertical angles the non-adjacent angles formed by intersecting lines. Vertical angles have the same measure. ©Curriculum Associates, LLC Copying is not permitted. Lesson 21 Understand Angle Relationships 219 Solve. l l 4 In the diagram, EA ' EC , and BD is a straight line. · · B Find the value of x. Show your work. E (3x 2 10)8 x8 A C D Solution: Use the diagram at the right to solve problems 5–6. 5 Explain how you would find m STQ. N M S (2x 1 8 )8 (71 2 x)8 T P Q 6 Find m STQ. Show your work. Solution: 220 Lesson 21 Understand Angle Relationships ©Curriculum Associates, LLC Copying is not permitted. Lesson 21 Name: Identify Angle Pairs Study the example showing how to use angle pair relationships to find measures of angles. Then solve problems 1–5. Example In the diagram, lines w, x, and y are parallel. Use the terms corresponding angles, alternate interior angles, and linear pair to describe some of the angle relationships in the diagram. w x 1 3 2 4 1 and 9 are corresponding angles. Corresponding angles formed by a transversal crossing parallel lines are congruent, so 1 > 9. 5 7 6 8 y 9 11 10 12 4 and 5 are alternate interior angles. Alternate interior angles between parallel lines are congruent, so 4 > 5. 1 and 3 form a linear pair. Linear pairs are supplementary, so m1 1 m 3 5 180°. 1 Use the diagram in the example. Name a different pair of corresponding angles, a different pair of alternate interior angles, and a different linear pair. corresponding angles: alternate interior angles: linear pair: 2 Use the diagram in the example. If m7 5 108°, what is m12? Explain how you found the measure. ©Curriculum Associates, LLC Copying is not permitted. Vocabulary transversal a line that crosses two lines. corresponding angles angles in the same position in the intersections of a transversal and two or more parallel lines. alternate interior angles angles between two parallel lines and on opposite sides of a transversal. linear pair adjacent supplementary angles. Lesson 21 Understand Angle Relationships 221 Solve. Use the diagram shown to solve problems 3–5. 3 In the diagram, lines a and b are parallel, with m 3 5 65° and m11 5 100°. Find the measures of 2, 5, and 13. Tell which angle relationships you used to help you find each measure. m 2: d c 1 2 4 3 a b 13 14 16 15 5 6 8 7 9 10 12 11 m 5: m13: 4 Find the measures of angles 3, 8, 9, and 14 to show that the sum of the interior angles of a trapezoid is 360°. Tell which angle relationships you used. Show your work. Solution: 5 Are 3 and 5 congruent? Explain. 222 Lesson 21 Understand Angle Relationships ©Curriculum Associates, LLC Copying is not permitted. Lesson 21 Name: Reason and Write Study the example. Underline two parts that you think make it a particularly good answer and a helpful example. Example Elena drew a map of a neighborhood park. She says that the park has three straight sidewalks, two of which are parallel. Her map shows vertical angles, corresponding angles, alternate interior angles, and linear pairs of angles that are formed by the three straight sidewalks in the park. Draw a map of a park that matches Elena’s description. Label the sidewalks m, n, and p, and label the angles formed by the sidewalks 1, 2, 3, 4, and so on. Describe how the lines for the sidewalks are related. Name one pair of angles in your map to illustrate each angle relationship mentioned in Elena’s description. Tell how you know that the angles illustrate the relationship. Show your work. Use a diagram, words, and angle relationships to explain your answer. Possible answer: m Where does the example . . . n 1 2 4 3 • answer each part of the problem? 5 6 8 7 p • use a diagram to illustrate? • use words to explain? Lines m and n are parallel, and line p is a transversal that intersects the two parallel lines. • use angle relationships to explain? /1 and /3 are vertical angles because they are non-adjacent angles formed by intersecting lines. /1 and /5 are corresponding angles because they are in the same position in the intersections of the transversal and the parallel lines. /2 and /8 are alternate interior angles because they are between two parallel lines and on opposite sides of the transversal. /1 and /2 form a linear pair because they are adjacent supplementary angles. ©Curriculum Associates, LLC Copying is not permitted. Lesson 21 Understand Angle Relationships 223 Solve the problem. Use what you learned from the model. Von makes a design with four lines. At least two of them are parallel. He says that his design shows vertical angles, corresponding angles, alternate interior angles, and linear pairs of angles. Draw a diagram of Von’s design. Label the lines a, b, c, and d, and label the angles formed by the lines 1, 2, 3, 4, and so on. Describe how the lines are related. List all of the angles that are congruent to 1 in your diagram. Then list all of the angles that are congruent to 2. Name one pair of angles in your diagram to illustrate each angle relationship mentioned in Von’s description. Tell how you know that the angles illustrate the relationship. Show your work. Use a diagram, words, and angle relationships to explain your answer. Did you . . . • answer each part of the problem? • use a diagram to illustrate? • use words to explain? • use angle relationships to explain? 224 Lesson 21 Understand Angle Relationships ©Curriculum Associates, LLC Copying is not permitted.