vertical angles Two adjacent angles are a linear pair if their noncommon sides are on the same line. 1 Identify Vertical Angles and Linear Pairs
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1 Page of 7. 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 Determine whether the labeled angles are vertical angles, a linear pair, or neither. a. b. c. Identify Vertical Angles and Linear Pairs 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. 5 6 c.a5 and a6 are vertical angles because they are not adjacent and their sides are formed by two intersecting lines. POSTULAT 7 Linear Pair Postulate Words If two angles form a linear pair, then they are supplementary. Symbols ma ma 80. Vertical Angles 75
2 Page of 7 Find the measure of arsu. Use the Linear Pair Postulate 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 R S U 6 T Student Help VISUAL STRATGY Draw an example of this theorem with specific measures, as shown on p. 5. THORM. Vertical Angles Theorem Words Vertical angles are congruent. Symbols aca and aca. 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 BAK To review the ongruent Supplements Theorem, see p. 69. a and a are supplementary to the same angle, so a is congruent to a by the ongruent Supplements Theorem. Use the Vertical Angles Theorem Find the measure of ad. A 50 D aab and ad are vertical angles. By the Vertical Angles Theorem, adcaab, so mad maab 50. B 76 hapter Segments and Angles
3 Page of 7 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. 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 Find the value of y. 5 Use Algebra with Vertical Angles Because the two expressions are measures of vertical angles, you can write the following equation. (y ) y Vertical Angles Theorem y y y y y y (y ) Subtract y from each side. Simplify. 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
4 Page of 7. xercises Guided Practice Vocabulary heck omplete 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 heck Find the measure of the numbered angle Find ma, ma, and ma Find the value of x (x 8) 6x x Practice and Applications xtra Practice See p Homework Help xample : xs. 9 xample : xs. 5 9 xample : xs. 0 xample : xs. 8 7 xample 5: xs 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 hapter Segments and Angles
5 Page 5 of 7 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 valuating 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. IStudent Help I L A S S Z O N. O M HOMWORK HLP xtra help with problem solving in xs. 8 is at classzone.com Finding Angle Measures Find ma, ma, and ma Flags ach flag shown contains vertical angles. Find ma, ma, and ma Scotland Dominican Republic Jamaica. Vertical Angles 79
6 Page 6 of 7 areers 7. Drafting Table The legs of the drafting table form vertical angles. Find the measures of a, a, and a. 85 RGONOMISTS study work conditions to improve the safety, efficiency, and comfort of workers. Drafting tables are angled so people can work at them without injuring their backs. areer Links L A S S Z O N. O M Finding Angle Measures Find ma, ma, ma, and ma Vertical Angles Use the diagram to complete the statement..abgc?.aagbc? B.aAG c?.agc? 50 G 5. maagf? 6. madg? A D F 7. mag? 8. mabg? 9. madgf? 50. maagd? 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 maab A 56. (6x 9) A B D B x 8n (9n 5) D A (5p 8) D p B 80 hapter Segments and Angles
7 Page 7 of hallenge 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,, ,,,,... ongruent Segments Determine which segments in the coordinate plane are congruent. (Lesson.5) 6. y 65. A(, ) (, ) J(, ) y K(, ) (, ) D(, ) L(, ) M(, ) B(, 0) G(, ) x F (, ) H(, ) Q(, ) N(, 0) R (0, ) P(, ) x Algebra Skills Simplifying xpressions 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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