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1-6 1-6 Measuring Angles 1. Plan Objectives 1 To find the measures of angles To identify special angle pairs 2 What You’ll Learn Check Skills You’ll Need • To find the measures of Solve each equation. angles • To identify special angle Examples 1 2 Naming Angles Measuring and Classifying Angles Using the Angle Addition Postulate Identifying Angle Pairs Making Conclusions From a Diagram 3 4 5 pairs . . . And Why 1. 50 1 a 5 130 80 2. m 2 110 5 20 130 3. 85 2 n 5 40 45 4. x 1 45 5 180 135 5. z 2 20 5 90 110 6. 180 2 y 5 135 45 Finding Angle Measures Math Background An angle (&) is formed by two rays with the same endpoint. The rays are the sides of the angle. The endpoint is the vertex of ) ) the angle. The sides of the angle shown here are BT and BQ . The vertex is B. You could name this angle &B, &TBQ, &QBT, or &1. Vocabulary Tip Angle measure and segment measure have important similarities. Congruent angles can be moved onto one another so they match exactly, as with congruent segments. Congruent angles are also indicated with tick marks. Like a ruler, the intervals on a protractor must be equal. The most common unit of angle measure is the degree, which results when a circle is divided into 360 equal parts. Finally, the Angle Addition Postulate corresponds directly with the Segment Addition Postulate. You may also refer to the angle suggested by the two segments BT and BQ as &TBQ. 1 EXAMPLE B T 1 Q Naming Angles Name &1 in two other ways. A C 1 2 E More Math Background: p. 2D Lesson Planning and Resources D &AEC and &CEA are other names for &1. Quick Check See p. 2E for a list of the resources that support this lesson. PowerPoint Bell Ringer Practice Check Skills You’ll Need For intervention, direct students to: Skills Handbook page 758 New Vocabulary • angle • acute angle • right angle • obtuse angle • straight angle • congruent angles • vertical angles • complementary angles • supplementary angles To find the measures of angles in flower arrangements, as in Exercise 41. 1 GO for Help 36 1 a. Name &CED two other ways. l2, lDEC b. Critical Thinking Would it be correct to name any of the angles &E? Explain. No; 3 ' have E for a vertex, so you need more info. in the name to distinguish them from one another. One way to measure an angle is in degrees. To indicate the size or degree measure of an angle, write a lowercase m in front of the 80⬚ angle symbol. The degree measure of angle A is 80. You show this A by writing m&A = 80. Chapter 1 Tools of Geometry Solving Linear Equations Skills Handbook, p. 758 Special Needs Below Level L1 Students with fine motor difficulties may have problems using a protractor. You may want to have students work in pairs so they can assist each other as needed. 36 learning style: verbal L2 Demonstrate on an overhead projector how to line up a protractor with a ray, and how to choose the appropriate scale. Have students choose the scale for which one of the sides passes through zero. learning style: visual Key Concepts Postulate 1-7 ) ) Let OA and OB be opposite ) ) rays in a plane. OA , OB , and all the rays with endpoint O that can be drawn on one * ) side of AB can be paired with the real numbers from 0 to 180 so that ) 2. Teach Protractor Postulate Guided Instruction 1 ) Error Prevention Discuss as a class why it is inappropriate to name &1 as &E. Ask: How could this cause confusion? There are three angles whose vertex is E. Explain also that the measure of an angle does not need a degree symbol. a. OA is paired with 0 and OB is paired with 180. ) EXAMPLE ) b. If OC is paired with x and OD is paired with y, then m&COD = ux - yu. Math Tip You can classify angles according to their measures. x⬚ x⬚ acute angle 0 < x < 90 right angle x ⫽ 90 x⬚ x⬚ obtuse angle 90 < x < 180 straight angle x ⫽ 180 Note the special symbol that’s tucked into the corner of the right angle. When you see it, you know that the measure of the angle is 90. 2 nline EXAMPLE Ask: How is the Protractor Postulate like the Ruler Postulate in Lesson 1-5? Both pair numbers in a one-to-one correspondence with geometric objects and use absolute value to determine measurements. Auditory Learners Have students take turns with a partner explaining the Protractor Postulate in their own words. Measuring and Classifying Angles 2 Find the measure of each angle. Classify each as acute, right, obtuse, or straight. a. EXAMPLE Connection to Algebra Review the meaning of the inequality symbol. b. PowerPoint Additional Examples Visit: PHSchool.com Web Code: aue-0775 120, obtuse Quick Check 90, right 1 Name the angle below in four ways. 2 Find the measure of each angle. Classify each as acute, right, obtuse, or straight. a. 30; acute b. 90; right c. 140; obtuse C G Angles with the same measure are congruent angles. In other words, if m&1 = m&2, then &1 > &2. You can use these statements interchangeably. A &3, &G, &AGC, &CGA 1 2 Find the measure of each angle. Classify each as acute, right, obtuse, or straight. 2 Angles can be marked alike to show that they are congruent, as in this photograph of the Air Force Thunderbirds precision flying team. 1 2 Lesson 1-6 Measuring Angles Advanced Learners 3 37 m&1 ≠ 110, obtuse; m&2 ≠ 80, acute English Language Learners ELL L4 Have students write their first names using block letters. Then have them count the number of acute, right, obtuse, and congruent angles suggested by the letters. learning style: visual Show how the concept and notation for congruent angles is closely related to congruent segments. For example, if AB DE, then AB = DE. Likewise, if &1 &2, then m&1 = m&2. learning style: visual 37 3 EXAMPLE Visual Learners The Angle Addition Postulate is similar to the Segment Addition Postulate. Draw the figures for the Angle Addition Postulate on the board. Have students place other points in the interior of &AOC to reinforce the concept of interior. Key Concepts Postulate 1-8 Angle Addition Postulate If point B is in the interior of &AOC, then m&AOB + m&BOC = m&AOC. A PowerPoint Additional Examples If &AOC is a straight angle, then m&AOB + m&BOC = 180. B B O A C C O 3 Suppose that m&1 = 42 and m&ABC = 88. Find m&2. 3 A What is m&TSW if m&RST = 50 and m&RSW = 125? 1 B Using the Angle Addition Postulate EXAMPLE W 2 m&RST + m&TSW = m&RSW 50 + m&TSW = 125 Quick Check Teaching Tip Subtract 50 from each side. 3 If m&DEG = 145, find m&GEF. 35 G D . 2 1 Error Prevention! Students sometimes confuse complementary and supplementary angles. One ways to keep them straight is to remember that c comes before s in the alphabet, just as 90 comes before 180. Angle Addition Postulate Substitute. m&TSW = 75 Guided Instruction After students read the definition of vertical angles, ask: What is another way to define vertical angles? opposite angles formed by two intersecting lines S R C m&2 ≠ 46 E F Identifying Angle Pairs Some angle pairs that have special names. X X X X vertical angles Helvetica Condensed Times Roman Eurostile Extended MarkerFelt Thin Chapter 1 Tools of Geometry 2 4 3 1 two angles whose sides are opposite rays complementary angles 2 4 two coplanar angles with a common side, a common vertex, and no common interior points supplementary angles 105⬚ 50⬚ 3 2 1 38 adjacent angles 1 3 In each font, a capital X suggests vertical angles. 38 T A B 40⬚ 4 75⬚ two angles whose measures have sum 90 two angles whose measures have sum 180 Each angle is called the complement of the other. Each angle is called the supplement of the other. 4 EXAMPLE 4 Identifying Angle Pairs a. complementary &2 and &3 c. vertical &3 and &5 2 1 5 Connection to Language Arts Students are familiar with the word compliment. Point out that the word in this lesson has an e instead of an i. Ask students to find non mathematical contexts where complement and supplement are used. In the diagram identify pairs of numbered angles that are related as follows: b. supplementary &4 and &5; &3 and &4 EXAMPLE 3 4 PowerPoint Quick Check 4 a. Name two pairs of adjacent angles in the photo below. Answers may vary. Sample: lAFB and b. If m&EFD = 27, find m&AFD. 153 lBFC; lBFD and lDFE Additional Examples 4 2 3 1 4 When entering the roadway, turn and look for oncoming traffic regardless of what you see in the rear-view mirror. Name all pairs of angles in the diagram that are a. vertical l1 and l3; l2 and l4 b. supplementary l1 and l2; l2 and l3; l3 and l4; l4 and l1 c. complementary none Whether you draw a diagram or use a given diagram, you can make some conclusions directly from the diagrams. You can conclude that angles are • adjacent angles 5a. Yes; the congruent segments are marked. b. No; there are no markings. c. No; there are no markings. d. No; there are no markings. nline • adjacent supplementary angles 5 Use the diagram from Example 2. Which of the following can you conclude: &3 is a right angle, &1 and &5 are adjacent, &3 &5? l1 and l5 are adjacent. • vertical angles Unless there are marks that give this information, you cannot assume • angles or segments are congruent • an angle is a right angle • lines are parallel or perpendicular 5 EXAMPLE Resources • Daily Notetaking Guide 1-6 L3 • Daily Notetaking Guide 1-6— L1 Adapted Instruction Making Conclusions From a Diagram What can you conclude from the information in the diagram? • &1 > &2, by the markings. 4 2 1 5 • &2 and &3, for example, are adjacent angles. Visit: PHSchool.com Web Code: aue-0775 3 Closure • &4 and &5, for example, are adjacent supplementary angles, or m&4 + m&5 = 180 by the Angle Addition Postulate. Ask: How are angles classified? By their angle measure: acute (R 90), right (≠ 90), obtuse (S 90), and straight (≠ 180) What are the special angle pairs? vertical, adjacent, complementary, and supplementary • &1 and &4, for example, are vertical angles. Quick Check 5 Can you make each conclusion from the information in the diagram? Explain. a. TW > WV b. PW > WQ a–d. See left. c. TV ' PQ d. TV bisects PQ. e. W is the midpoint of TV.Yes; the congruent segments are marked. T P W Q V Lesson 1-6 Measuring Angles 39 39 EXERCISES 3. Practice For more exercises, see Extra Skill, Word Problem, and Proof Practice. Practice and Problem Solving Assignment Guide A 1 A B 1-14, 33-47 2 A B Practice by Example Example 1 15-32 C Challenge 48-49 Test Prep Mixed Review 50-54 55-59 GO for Help Name each angle in three ways. 1. Example 2 (page 37) Exercises 9–12 Using a corner of paper to model a 90° angle makes classifying acute and obtuse angles visually apparent. GPS Guided Problem Solving 7. a straight angle, &EFG 8. a right angle, &GHI about 42° 10. the angle formed by the skis 11. L3 90; right B Example 4 L3 Date (page 39) The Coordinate Plane Graph each point in the coordinate plane. 7. N(1, 0), P(3, 8) 9. S(0, 5), T(0, ⫺3) 14. E(14, ⫺2), F(7, ⫺8) 15. O(0, 0), G(⫺5, 12) 16. H(2.8, 1.1), I(⫺3.4, 5.7) 17. J(2 12 , - 14 ), K(3 14 , -1) 24. What is the perimeter of PQSR? 25. What is the midpoint of QR? 40 © Pearson Education, Inc. All rights reserved. 23. Graph quadrilateral PQSR. E B 60⬚ O D C In the diagram above, find the measure of each of the following angles. 20. The midpoint of EF is (⫺3, 7). The coordinates of E are (⫺3, 10). Find the coordinates of F. Quadrilateral PQSR has coordinates as follows: P(0, 0), Q(–1, 4), R(8, 2), and S(7, 6). A 19. a pair of vertical angles lAOB and lDOC or lBOC and lAOD 19. The midpoint of CD is (4, 11). The coordinates of D are (4, 12). Find the coordinates of C. 22. A crow flies to a point that is 1 mile east and 20 miles south of its starting point. How far does the crow fly? 15. supplementary to &AOD lAOB or lDOC 18. complementary to &EOD lDOC or lAOB 18. The midpoint of AB is (1, 2). The coordinates of A are (⫺3, 6). Find the coordinates of B. 21. Graph the points A(2, 1), B(2, ⫺5), C(⫺4, ⫺5), and D(⫺4, 1). Draw the segments connecting A, B, C, and D in order. Are the lengths of the sides of ABCD the same? Explain. E Name an angle or angles in the diagram described by each of the following. 17. supplementary to &EOA lEOC 11. W(2, 7), X(1, 2) 13. C(⫺1, 5), D(2, ⫺3) F 16. adjacent and congruent to &AOE lEOC Find the coordinates of the midpoint of each segment. The coordinates of the endpoints are given. 12. A(6, 7), B(4, 3) J A 5. E(⫺4, ⫺2) Find the distance between the points to the nearest tenth. 8. Q(10, 10), R(10, ⫺2) 14. Find m&GFJ if m&EFG = 110. 70 G L1 Adapted Practice 10. U(11, 0), V(⫺1, 0) 13. Find m&CBD if m&ABC = 45 and m&ABD = 79. 34 C L2 Reteaching 135; obtuse D L4 Enrichment 6. L(⫺4, 11), M(⫺3, 4) D 60; acute (page 38) 4. D(⫺4, 0) 1 2 B A 5–8. See margin. 6. an acute acute, &BCD 9. Example 3 3. C(0, 6) C 5. an obtuse angle, &RST 12. 2. B(5, ⫺2) lMCP, lPCM, lC, or l1 Use a protractor. Measure and classify each angle. Exercise 16 Point out to students that two conditions must be met in this exercise. 1. A(⫺2, 5) Z 4. &2 lCBD, lDBC Draw and label a figure to fit each description. Error Prevention! Class P M 3. &1 lABC, lCBA Tactile Learners Practice 1-6 C 1 Use the figure at the right. Name the indicated angle in two different ways. To check students’ understanding of key skills and concepts, go over Exercises 12, 18, 33, 41, 47. Name lXYZ, lZYX, lY Y Homework Quick Check Practice 2. X (page 36) 20. &EOC 90 40 Chapter 1 Tools of Geometry 21. &DOC 30 22. &BOC 150 23. &AOB 30 (page 39) 24. &J > &D Yes; the markings show they are congruent. 25. &JAC > &DAC No; there are no markings. 26. &JAE and &EAF are adjacent and supplementary. See left. 26. Yes; you can conclude that the angles are adjacent and supplementary from the diagram. 28. Yes; you can conclude that angles are supplementary from the diagram. B Exercises 24, 25 Ask: Why are three letterers used to name the angles in Exercise 25 but only one letter is used in Exercise 24? Vertex J and vertex D each apply to only one angle, but many angles share vertex A. Can you make each conclusion from the information in the diagram? Explain. Example 5 E F A 27. m&JCA = m&DCA No; there are no markings. 28. m&JCA + m&ACD = 180 See left. C J D Error Prevention! No; there are no markings. 29. AJ > AD 30. C is the midpoint of JD. Yes; there are markings. ) 31. &EAF and &JAD are vertical angles. 32. AC bisects &JAD. No; there are no markings. See left. A In the diagram, mlACB ≠ 65. Find each of the following. Apply Your Skills 33. m&BCD 115 31. Yes; you can conclude that ' are vertical from the diagram. 34. m&ECD 65 E 35. 6:00 180 36. 7:00 150 37. 11:00 30 38. 4:40 100 39. 5:20 40 40. 10:40 80 B C Estimation Estimate the measure of the angle formed by the hands of a clock at each time. D Exercises 33–34 41. Flower Arranging In Japanese flower arranging, you match a stem that is vertical with 0. You match other stems with numbers from 0 to 90, in both directions from the vertical. What numbers would the flowers shown be paired with on a standard protractor? 45, 75, and 165, or 135, 105, and 15 Real-World 42. 12; mlAOC ≠ 82, mlAOB ≠ 32; mlBOC ≠ 50 43. 8; mlAOB ≠ 30, mlBOC ≠ 50; mlCOD ≠ 30 44. 18; mlAOB ≠ 28, ml BOC ≠ 52; mlAOD ≠ 108 x 2 Algebra Use the diagram, below right, for Exercises 42–45. Solve for x. Find the Connection angle measures to check your work. 42–45. See margin. Japanese flower arranging makes precise use of angles to create a mood. Exercise 46 Students may be misled or confused because the drawing is not drawn to scale with m&MQV = 90. Students can redraw the figure, but by examining the answer choices students should be able to identify the correct answer. 42. m&AOC = 7x - 2, m&AOB = 2x + 8, m&BOC = 3x + 14 45. 7; mlAOB ≠ 31, mlBOC ≠ 49; mlAOD ≠ 111 B A C 43. m&AOB = 4x - 2, m&BOC = 5x + 10, m&COD = 2x + 14 O D 44. m&AOB = 28, m&BOC = 3x - 2, m&AOD = 6x 45. m&AOB = 4x + 3, m&BOC = 7x, m&AOD = 16x - 1 46. Multiple Choice If m&MQV = 90, which expression can you use to find m&VQP? A m/MQP 2 90 90 2 m/MQV m/MQP 1 90 90 1 m/VQP 47c. Answers may vary. Sample: The sum of the l measures should be 180. GO Homework Help GPS Visit: PHSchool.com Web Code: aua-0106 and m&TQS = 6x + 20. 19.5 b. What is m&RQS? m&TQS? 43; 137 c. Show how you can check your answer. See left. M N 6. R 8. B Q T 41 G C 5. D T S S Lesson 1-6 Measuring Angles lesson quiz, PHSchool.com, Web Code: aua-0106 5–8. Drawings may vary. Samples are given. P V x 2 47. a. Algebra Solve for x if m&RQS = 2x + 4 nline R Q 7. E F H I G 41 4. Assess & Reteach C Challenge PowerPoint ) ) ) ) 48. XC bisects &AXB, XD bisects &AXC, ) ) XE bisects &AXD, XF bisects &EXD, XG bisects &EXF, and XH bisects &DXB. If m&DXC = 16, find m&GXH. 30 49. Technology Leon constructed an angle. Then he constructed a ray from the vertex of the angle to a point in the interior of the angle. He measured all the angles formed. Then he moved the interior ray. What postulate do the two pictures support? Angle Add. Post. Lesson Quiz Use the figure below for Exercises 1–2. C D 105⬚ 105⬚ 1 A 2 B 24⬚ 1. Name &2 two different ways. &DAB, &BAD 2. Measure and classify &1, &2, and &BAC. 90, right; 30, acute; 120, obtuse 2 42⬚ Test Prep Use the figure below for Exercises 3–4. 1 63⬚ 81⬚ Multiple Choice 3 4 50. Two angles are congruent, adjacent, and supplementary. What is the measure of each? B A. 45 B. 90 C. 180 D. cannot be determined 51. Two angles are congruent and complementary. What is the measure of each? F F. 45 G. 90 H. 180 J. cannot be determined 52. Two angles are adjacent and supplementary. What is the measure of each? D A. 45 B. 90 C. 180 D. cannot be determined 3. Name a pair of supplementary angles. Samples: l1 and l3, l2 and l4 53. When 15 is subtracted from the measure of an angle, the result is the measure of a right angle. What is the measure of the original angle? H F. 75 G. 85 H. 105 J. 115 4. Can you conclude that there are vertical angles in the diagram? Explain. No; no angle pairs are formed by opposite rays. Short Response 54. You are given that m&ABD + m&DBC = m&ABC. a. Draw a diagram to show the above. a–b. See margin. b. If m&ABD = 12 and &ABC is obtuse, what are the least and greatest whole number measures possible for &DBC? Explain. Alternative Assessment Have students draw diagrams to illustrate the Angle Addition Postulate. Then have them write examples that use each postulate to find a missing measurement when two of the three measurements are known. Mixed Review Lesson 1-5 GO for Help Test Prep 55. If EG 5 75 and EF 5 28, what is FG? 47 E 56. If EG 5 49, EF 5 2x 1 3, and FG 5 4x 2 2, find x. Then find EF and FG. x = 8; EF = 19; FG = 30 42 57. Writing Explain the difference between an orthographic drawing and an isometric drawing. See back of book. Lesson 1-1 Find one counterexample to show that each conjecture is false. 58. The quotient of two integers is not an integer. 452 2 Chapter 1 Tools of Geometry 54. [2] a. D A C B 42 F Lesson 1-2 Resources For additional practice with a variety of test item formats: • Standardized Test Prep, p. 75 • Test-Taking Strategies, p.70 • Test-Taking Strategies with Transparencies Use the figure at the right for Exercises 55–56. b. An obtuse l measures between 90 and 180 degrees; the least and greatest whole number values are 91 and 179 degrees. Part of lABC is 12°. So the least and greatest l measures for lDBC are 79 and 167. 59. An even number cannot have 5 as a factor. 10 = 2 ⫻ 5, so 10 has 5 as a factor and 10 is even. [1] one part correct G