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NAME ______________________________________________ DATE 1-4 ____________ PERIOD _____ Study Guide and Intervention Angle Measure Measure Angles If two noncollinear rays have a common endpoint, they form an angle. The rays are the sides of the angle. The common endpoint is the vertex. The angle at the right can be named as /A, /BAC, /CAB, or /1. B 1 A A right angle is an angle whose measure is 90. An acute angle has measure less than 90. An obtuse angle has measure greater than 90 but less than 180. Example 1 S R 1 2 C Example 2 Measure each angle and classify it as right, acute, or obtuse. T 3 Q P E D a. Name all angles that have R as a vertex. Three angles are /1, /2, and /3. For other angles, use three letters to name them: /SRQ, /PRT, and /SRT. A B C a. /ABD Using a protractor, m/ABD 5 50. 50 , 90, so /ABD is an acute angle. b. Name the sides of /1. ##$, RP ##$ RS b. /DBC Using a protractor, m/DBC 5 115. 180 . 115 . 90, so /DBC is an obtuse angle. c. /EBC Using a protractor, m/EBC 5 90. /EBC is a right angle. Exercises A B 4 1. Name the vertex of /4. 1 D 2. Name the sides of /BDC. 3 2 C 3. Write another name for /DBC. Measure each angle in the figure and classify it as right, acute, or obtuse. N M S 4. /MPR P 5. /RPN R 6. /NPS © Glencoe/McGraw-Hill 19 Glencoe Geometry Lesson 1-4 Refer to the figure. NAME ______________________________________________ DATE 1-4 ____________ PERIOD _____ Study Guide and Intervention (continued) Angle Measure Congruent Angles Angles that have the same measure are congruent angles. A ray that divides an angle into two congruent angles is called an angle bisector. In the figure, ##$ PN is the angle bisector of /MPR. Point N lies in the interior of /MPR and /MPN > /NPR. M N P R Q R Example Refer to the figure above. If m/MPN 5 2x 1 14 and m/NPR 5 x 1 34, find x and find m/MPR. Since ##$ PN bisects /MPR, /MPN > /NPR, or m/MPN 5 m/NPR. 2x 1 14 5 x 1 34 2x 1 14 2 x 5 x 1 34 2 x x 1 14 5 34 x 1 14 2 14 5 34 2 14 x 5 20 m/NPR 5 (2x 1 14) 1 (x 1 34) 5 54 1 54 5 108 Exercises ##$ bisects /PQT, and QP ##$ and QR ##$ are opposite rays. QS 1. If m/PQT 5 60 and m/PQS 5 4x 1 14, find the value of x. S T P 2. If m/PQS 5 3x 1 13 and m/SQT 5 6x 2 2, find m/PQT. ##$ and BC ##$ are opposite rays, BF ##$ bisects /CBE, and BA #BD #$ bisects /ABE. E D 3. If m/EBF 5 6x 1 4 and m/CBF 5 7x 2 2, find m/EBC. F 1 A 2 3 B 4 C 4. If m/1 5 4x 1 10 and m/2 5 5x, find m/2. 5. If m/2 5 6y 1 2 and m/1 5 8y 2 14, find m/ABE. 6. Is /DBF a right angle? Explain. © Glencoe/McGraw-Hill 20 Glencoe Geometry NAME ______________________________________________ DATE 1-5 ____________ PERIOD _____ Study Guide and Intervention Angle Relationships Pairs of Angles Adjacent angles are angles in the same plane that have a common vertex and a common side, but no common interior points. Vertical angles are two nonadjacent angles formed by two intersecting lines. A pair of adjacent angles whose noncommon sides are opposite rays is called a linear pair. Example Identify each pair of angles as adjacent angles, vertical angles, and/or as a linear pair. a. b. S T U R M 4 R /SRT and /TRU have a common vertex and a common side, but no common interior points. They are adjacent angles. c. A S /1 and /3 are nonadjacent angles formed by two intersecting lines. They are vertical angles. /2 and /4 are also vertical angles. 608 6 B 3N 2 d. D 5 P 1 C 308 /6 and /5 are adjacent angles whose noncommon sides are opposite rays. The angles form a linear pair. B A 1208 F 608 G /A and /B are two angles whose measures have a sum of 90. They are complementary. /F and /G are two angles whose measures have a sum of 180. They are supplementary. Exercises Identify each pair of angles as adjacent, vertical, and/or as a linear pair. 2. /1 and /6 V 2 1 3. /1 and /5 4. /3 and /2 3 4 6Q R R S P For Exercises 5–7, refer to the figure at the right. 5. Identify two obtuse vertical angles. S 5 V N U 6. Identify two acute adjacent angles. T 7. Identify an angle supplementary to /TNU. 8. Find the measures of two complementary angles if the difference in their measures is 18. © Glencoe/McGraw-Hill 25 Glencoe Geometry Lesson 1-5 1. /1 and /2 T U NAME ______________________________________________ DATE 1-5 ____________ PERIOD _____ Study Guide and Intervention (continued) Angle Relationships Perpendicular Lines Lines, rays, and segments that form four right angles are perpendicular. The right angle symbol indicates that the lines #$ is perpendicular to @#$ are perpendicular. In the figure at the right, @AC BD , @ #$ @#$ or AC ⊥ BD . A B C D Example Find x so that D wZ w⊥w PZ w. If D wZ w⊥P wZ w, then m/DZP 5 90. m/DZQ 1 m/QZP (9x 1 5) 1 (3x 1 1) 12x 1 6 12x x 5 5 5 5 5 m/DZP 90 90 84 7 D Q (9x 1 5)8 (3x 1 1)8 Sum of parts 5 whole Substitution Z Simplify. P Subtract 6 from each side. Divide each side by 12. Exercises #$ ⊥ @MQ #$. 1. Find x and y so that @NR N P 2. Find m/MSN. 5x 8 M x8 (9y 1 18)8 S Q R #$ ⊥ #BF #$. Find x. 3. m/EBF 5 3x 1 10, m/DBE 5 x, and #BD E D 4. If m/EBF 5 7y 2 3 and m/FBC 5 3y 1 3, find y so #$ ⊥ ##$ that #EB BC . F B A C 5. Find x, m/PQS, and m/SQR. P S 3x 8 (8x 1 2)8 Q R 6. Find y, m/RPT, and m/TPW. T (4y 2 5)8 (2y 1 5)8 R P W V S © Glencoe/McGraw-Hill 26 Glencoe Geometry