Activity. Angles and Intersecting Lines. Question. Materials. Explore. Think About It. Student Help
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1 Activity. Angles and Intersecting Lines Question What is the relationship between the angles formed by two intersecting lines? Materials straightedge protractor Explore On a piece of paper, draw line l using a straightedge. Label two points A and B on the line. Draw line m so that it intersects line l. Label the point of intersection E. Label two points C and D on line m as shown below. Use a protractor to measure the four angles formed by the intersecting lines. Record the angle measures. Student Help VOCABULARY TIP Recall from Lesson. that adjacent angles share a common vertex and a common side. a and a are adjacent angles. Think About It. What do you notice about the nonadjacent angles you measured in Step?. Find the sum of the measures of any two adjacent angles in Step. What do you notice?. Repeat Steps using two different lines. What do you notice about the measures of adjacent angles and nonadjacent angles?. Extension Draw a third line, n, that goes through point E. Use a protractor to measure the six angles formed by the intersecting lines. Record your results. What do you notice about the angle measures? 7 Chapter Segments and Angles
2 . Vertical Angles Goal Find the measures of angles formed by intersecting lines. Key Words vertical angles linear pair Two angles are vertical angles if they are not adjacent and their sides are formed by two intersecting lines. The scissors show two sets of vertical angles. a and a are vertical angles. a and a are vertical angles. Two adjacent angles are a linear pair if their noncommon sides are on the same line. a5 and a6 are a linear pair. noncommon side common side 5 6 noncommon side Visualize It! You can use colored pencils to help you see pairs of vertical angles. 6 5 Vertical angles a and a a and a5 a and a6 EXAMPLE Identify Vertical Angles and Linear Pairs Determine whether the labeled angles are vertical angles, a linear pair, or neither. a. b. c. 5 6 a. a and a are a linear pair because they are adjacent and their noncommon sides are on the same line. b. a and a are neither vertical angles nor a linear pair. c. a5 and a6 are vertical angles because they are not adjacent and their sides are formed by two intersecting lines. POSTULATE 7 Linear Pair Postulate Words If two angles form a linear pair, then they are supplementary. Symbols ma ma 80. Vertical Angles 75
3 EXAMPLE Use the Linear Pair Postulate Find the measure of arsu. 6 R S T arsu and aust are a linear pair. By the Linear Pair Postulate, they are supplementary. To find marsu, subtract maust from 80. marsu 80 maust U Student Help VISUAL STRATEGY Draw an example of this theorem with specific measures, as shown on p. 5. THEOREM. Vertical Angles Theorem Words Vertical angles are congruent. Symbols a c a and a c a. The following steps show why the Vertical Angles Theorem is true. a and a are a linear pair, so a and a are supplementary. Student Help a and a are a linear pair, so a and a are supplementary. LOOK BACK To review the Congruent Supplements Theorem, see p. 69. a and a are supplementary to the same angle, so a is congruent to a by the Congruent Supplements Theorem. EXAMPLE Use the Vertical Angles Theorem Find the measure of aced. A 50 E D C B aaeb and aced are vertical angles. By the Vertical Angles Theorem, aced caaeb, so maced maaeb Chapter Segments and Angles
4 Student Help STUDY TIP When you know the measure of one vertical angle, an easy starting point is to fill in the measure of the other. EXAMPLE Find Angle Measures Find ma, ma, and ma. 5 ma 5 Vertical Angles Theorem ma Linear Pair Postulate ma ma 5 Vertical Angles Theorem Find Angle Measures Find ma, ma, and ma EXAMPLE 5 Use Algebra with Vertical Angles Find the value of y. Because the two expressions are measures of vertical angles, you can write the following equation. (y ) y Vertical Angles Theorem y y y y Subtract y from each side. y Simplify. y (y ) y Simplify. y Divide each side by. Use Algebra with Angle Measures Find the value of the variable (r ) 89 x (x 6) (0t 5) 5t. Vertical Angles 77
5 . Exercises Guided Practice Vocabulary Check Complete the statement.. Two adjacent angles whose noncommon sides are on the same line are called?.. Two angles are called? if they are not adjacent and their sides are formed by two intersecting lines. Skill Check Find the measure of the numbered angle Find ma, ma, and ma Find the value of x (x 8) 6x x Practice and Applications Extra Practice See p Homework H lp Example : Exs. 9 Example : Exs. 5 9 Example : Exs. 0 Example : Exs. 8 7 Example 5: Exs Vertical Angles and Linear Pairs Determine whether the angles are vertical angles, a linear pair, or neither. 9. a5 and a6 0. a5 and a9. a5 and a8. a6 and a9. a8 and a9. a5 and a7 Using the Linear Pair Postulate Find the measure of a Chapter Segments and Angles
6 Linear Pairs Find the measure of the angle described. 8. a and a are a linear pair, and ma 5. Find ma. 9. a and a are a linear pair, and ma. Find ma. Using the Vertical Angles Theorem Find the measure of a Evaluating Statements Use the figure below to decide whether the statement is true or false.. If ma 0, then ma 0.. If ma 0, then ma a and a are a linear pair. 6. ma ma ma ma 7. a and a are vertical angles. Student Help I CLASSZONE.COM HOMEWORK HELP Extra help with problem solving in Exs. 8 is at classzone.com Finding Angle Measures Find ma, ma, and ma Flags Each flag shown contains vertical angles. Find ma, ma, and ma Scotland Dominican Republic Jamaica. Vertical Angles 79
7 Careers 7. Drafting Table The legs of the drafting table form vertical angles. Find the measures of a, a, and a. 85 ERGONOMISTS study work conditions to improve the safety, efficiency, and comfort of workers. Office furniture is designed so that people can work with minimal physical strain. Career Links CLASSZONE.COM Finding Angle Measures Find ma, ma, ma, and ma Vertical Angles Use the diagram to complete the statement.. abgc c?. aagb c? B. aagc c?. acge c? 50 G 5. maagf? 6. madge? A F 7. macge? 8. mabgc? 9. madgf? 50. maagd? E C D Using Algebra Find the value of the variable (w 7) 78 (5y ) 05 (x ) Using Algebra Find the value of the variable. Then use substitution to find maabc A 56. E (6x 9) C A B D E B x 8n (9n 5) D C C A (5p 8) D p B E 80 Chapter Segments and Angles
8 57. Challenge Find the values of x and y in the diagram below. (7x ) 8y (x ) 58. Visualize It! Sketch and label four angles so that a and a are acute vertical angles, a is a right angle adjacent to a, and a and a form a linear pair. Standardized Test Practice 59. Multi-Step Problem Use the diagram below. 6x (x 8) a. Use the Vertical Angles Theorem to write an equation. b. Solve your equation to find the value of x. c. Find the measures of the acute angles formed by the lines. d. Find the measures of the obtuse angles formed by the lines. Mixed Review Describing Number Patterns Describe a pattern in the numbers. Write the next number you expect in the pattern. (Lesson.) 60.,, 8, 5,... 6., 5, 75, 75,... 6., 6, 8,, ,,,,... Congruent Segments Determine which segments in the coordinate plane are congruent. (Lesson.5) 6. y 65. A(, ) E(, ) J(, ) y K(, ) C(, ) D(, ) L(, ) M(, ) B(, 0) G(, ) x F (, ) H(, ) Q(, ) N(, 0) R (0, ) P(, ) x Algebra Skills Simplifying Expressions Simplify the expression. (Skills Review, p. 67) 66. 6x 9x a a 68. 8z 5z 69. 6b 6b b 70. (t ) t 7. w w 5. Vertical Angles 8
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