Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Multilateration wikipedia , lookup
Golden ratio wikipedia , lookup
Noether's theorem wikipedia , lookup
Four color theorem wikipedia , lookup
Perceived visual angle wikipedia , lookup
Reuleaux triangle wikipedia , lookup
Euler angles wikipedia , lookup
History of trigonometry wikipedia , lookup
Rational trigonometry wikipedia , lookup
Trigonometric functions wikipedia , lookup
Euclidean geometry wikipedia , lookup
Parallel Lines and the Triangle Angle-Sum Theorem 3-4 3-4 1. Plan What You’ll Learn Check Skills You’ll Need • To classify triangles and find Classify each angle as acute, right, or obtuse. the measures of their angles GO for Help 1. 2. Lesson 1-6 Objectives 1 2 3. • To use exterior angles of triangles acute right . . . And Why acute Examples 1 Solve each equation. To find the reclining angle of a lounge chair, as in Example 4 4. 30 + 90 + x = 180 60 5. 55 + x + 105 = 180 20 6. x + 58 = 90 32 7. 32 + x = 90 58 2 3 New Vocabulary • acute triangle • right triangle • obtuse triangle • equiangular triangle • equilateral triangle • isosceles triangle • scalene triangle • exterior angle of a polygon • remote interior angles 1 4 The diagrams at the top of the Activity Lab on page 146 suggest the Triangle-Angle Sum Theorem. Theorem 3-12 Triangle Angle-Sum Theorem C A When alterations of Euclid’s Parallel Postulate lead to different geometries, the Triangle AngleSum Theorem appears strikingly different. In a hyperbolic geometry, the sum of a triangle’s angle measures is less than 180; in an elliptic geometry, the sum is greater than 180. More Math Background: p. 124C The sum of the measures of the angles of a triangle is 180. m&A + m&B + m&C = 180 Applying the Triangle Angle-Sum Theorem Classifying a Triangle Using the Exterior Angle Theorem Real-World Connection Math Background Finding Angle Measures in Triangles Key Concepts To classify triangles and find the measures of their angles To use exterior angles of triangles B Lesson Planning and Resources The following proof of Theorem 3-12 relies on the idea that through a point not on a given line you can draw a line parallel to the given line. Proof Proof of Theorem 3-12 See p. 124E for a list of the resources that support this lesson. PowerPoint Given: #ABC P C Bell Ringer Practice 13 2 Prove: m&A + m&B + m&3 = 180 Proof: By the Protractor Postulate, you can draw ) CP so that m&1 = m&A. Then, &1 and &A are * ) congruent alternate interior angles, so CP 6 AB. A &2 and &B are also alternate interior angles, so by the Alternate Interior Angles Theorem, m&2 = m&B. By substitution, m&A + m&B + m&3 = m&1 + m&2 + m&3, which is equal to 180 by the Angle Addition Postulate. Check Skills You’ll Need For intervention, direct students to: B Finding Angle Measures Lesson 1-6; Example 2 Extra Skills, Word Problems, Proof Practice, Ch. 1 Solving Linear Equations Algebra 1 Review, page 30 Lesson 3-4 Parallel Lines and the Triangle Angle-Sum Theorem Special Needs Below Level L1 Tell students that acute, right, and obtuse angles can help them identify acute, right, and obtuse triangles. Have students prepare a chart that shows each triangle with its defining characteristic in color. learning style: visual 147 L2 Using geometry software to draw, measure, and manipulate the seven triangles on page 148 will help students discover results such as the Isosceles Triangle Theorem. learning style: visual 147 2. Teach 1 Applying the Triangle Angle-Sum Theorem EXAMPLE Algebra Find the values of x and y. Guided Instruction 39 + 65 + x = 180 104 + x = 180 Each new lesson requires students to keep track of the names of more and more theorems, so shorthand ways to write their names can help students remember them. Point out that the symbol S is used in mathematics to indicate a sum. So, they could abbreviate the Triangle Angle-Sum Theorem as: x = 76 EXAMPLE Simplify. Subtract 104 from each side. m&GJF + m&GJH = 180 65 x y z H J F Angle Addition Postulate x + y = 180 76 + y = 180 y = 104 Quick Check Substitute. Substitute 76 for x. Subtract 76 from each side. 1 Find the value of z in two different ways, each way using the Triangle Angle-Sum Theorem. 55 Alternative Method Before finding any values, ask: How many triangles are in the diagram? 3 Use the question to highlight the alternate method of first finding the value of z using GFH. 2 39 Triangle Angle-Sum Theorem To find the value of y, look at &FJH. It is a straight angle. &S EXAMPLE 21 To find the value of x, use #GFJ. Teaching Tip 1 G In Chapter 1, you classified an angle by its measure. You can also classify a triangle by its angles and sides. nline Equiangular all angles congruent Math Tip Students have not yet learned that a triangle is equilateral if and only if it is equiangular. As students see more triangles, encourage them to develop hypotheses about triangles, including what is impossible, such as an equilateral obtuse triangle. Acute all angles acute Right one right angle Obtuse one obtuse angle Visit: PHSchool.com Web Code: aue-0775 Equilateral all sides congruent 2 EXAMPLE Isosceles at least two sides congruent Scalene no sides congruent Classifying a Triangle PowerPoint Classify the triangle by its sides and its angles. Additional Examples 1 In triangle ABC, &ACB is a right angle, and CD # AB. Find the values of a, b, and c. C c° A At least two sides are congruent, so the triangle is isosceles. All the angles are acute, so the triangle is acute. 70° b° a° D The triangle is an acute isosceles triangle. B Quick Check a = 70, b = 20, c = 20 2 Classify the triangle by its sides and its angles. 5 4 148 2 Draw and mark a triangle to fit each description. If no triangle can be drawn, write not possible and explain why. a–c. See margin. a. acute scalene b. isosceles right c. obtuse equiangular Chapter 3 Parallel and Perpendicular Lines 2 obtuse scalene Advanced Learners English Language Learners ELL L4 After Example 1, help students discover the Triangle Exterior Angle Theorem by finding x and y as sums of angles in the triangles. x21°55°, y65°39° 148 learning style: visual To help students remember the meaning of remote interior angle, use the word remote in other contexts, such as a remote island or a television remote control. learning style: verbal Guided Instruction 2 1 Using Exterior Angles of Triangles Tactile Learners An exterior angle of a polygon is an angle formed by a side and an extension of an adjacent side. For each exterior angle of a triangle, the two nonadjacent interior angles are its remote interior angles. Interior angle of a triangle means the same as “angle of a triangle.” 3 Encourage students to copy the triangle diagram shown immediately above the Triangle Exterior Angle Theorem. Have them cut out the exterior angle and two remote interior angles and then superimpose the interior angles over the exterior angle to demonstrate the Triangle Exterior Angle Theorem. 2 Remote interior angles The diagram at the right suggests a relationship between an exterior angle and its two remote interior angles. Theorem 3-13 states this relationship. You will prove this theorem in Exercise 35. 1 3 3 2 Vocabulary Tip Exterior angle 1 2 4 Key Concepts Theorem 3-13 Triangle Exterior Angle Theorem 2 The measure of each exterior angle of a triangle equals the sum of the measures of its two remote interior angles. 1 3 m&1 = m&2 + m&3 Connection to Algebra EXAMPLE Suggest that students let x represent the measure of the angle to help them associate the equations with more familiar algebraic equations. PowerPoint 3 EXAMPLE Using the Exterior Angle Theorem Algebra Find each missing angle measure. a. b. 40 1 30 m&1 = 40 + 30 For: Triangle Theorems Activity Use: Interactive Textbook, 3-4 Quick Check 5 A B B E 1 43 = m&2 35 3 Two angles of a triangle measure 45. Find the measure of an exterior angle at each vertex. 135, 135, 90 EXAMPLE E D D C 2 Real-World Connection 4 Explain what happens to the angle formed by the back of the chair and the armrest as you make a lounge chair recline more. The angle increases in measure. Resources • Daily Notetaking Guide 3-4 L3 • Daily Notetaking Guide 3-4— L1 Adapted Instruction E D C 125° 70 Multiple Choice The lounge chair has different settings that change the angles formed by its parts. Suppose m&2 is 32 and m&3 is 81. Find m&1, the angle formed by the back of the chair and the arm rest. E D C B A D C B A 4 C B A 3 C B A 2 3 Find m&1. 113 113 = 70 + m&2 m&1 = 70 4 1 Additional Examples D E E 67 113 Test-Taking Tip To check exterior angle-measure answers, remember that an exterior angle and the adjacent interior angle must be supplementary. 81 180 m&1 = m&2 + m&3 Exterior Angle Theorem m&1 = 32 + 81 Substitute. m&1 = 113 Simplify. 1 Closure 2 3 Explain what is wrong with this diagram. B The angle formed is a 1138 angle. The correct choice is C. Lesson 3-4 Parallel Lines and the Triangle Angle-Sum Theorem 149 C Quick Check b. 10. 2. a. c. Not possible; an equilateral k has all acute '. 11. Not possible; a right k will always have one longest side opp. the right l. 85° 80° A D m&BAD must be greater than m&BCA. 149 3. Practice Quick Check Assignment Guide 1 A B 1-15, 23-25, 27-32, 34, EXERCISES 37, 38 For more exercises, see Extra Skill, Word Problem, and Proof Practice. Practice and Problem Solving 2 A B 16-22, 26, 33, 35, 36 C Challenge 39-41 Test Prep Mixed Review 4 a. Change the setting on the lounge chair so that m&2 = 33 and m&3 = 97. Find the new measure of &1. 130 b. Explain how you can find m&1 without using the Exterior Angle Theorem. Answers may vary. Sample: Find the measure of the third l of the triangle. Subtract this from 180. A 42-44 45-48 Practice by Example Find ml1. Example 1 GO for Help Homework Quick Check 1. (page 148) 2. 117 1 33 30 3. 52.2 83.1 44.7 To check students’ understanding of key skills and concepts, go over Exercises 4, 18, 25, 28, 34. 33 57 1 90 1 x 2 Algebra Find the value of each variable. 4. Exercise 11 Discuss why this figure is impossible to draw. When students say that the side opposite the right angle is always longer than either of the other sides, point out that their observation is a theorem that the longest side of a triangle is always opposite the angle with the greatest measure. x 70; y 110; z 30 5. 30 70 6. 60 30 30 40 x 80 x y y z x 80; y 80 c Use a protractor and a centimeter ruler to measure the angles and the sides of each triangle. Classify each triangle by its angles and sides. Example 2 (page 148) 7. 8. 9. obtuse, isosceles acute, equiangular, equilateral right, scalene If possible, draw a triangle to fit each description. Mark the triangle to show known information. If no triangle can be drawn, write not possible and explain why. 10. acute equilateral 10–15. See margin 11. equilateral right 12. obtuse scalene pp. 150–151. 13. obtuse isosceles 14. isosceles right GPS Guided Problem Solving (page 149) L2 Reteaching b. l1 and l3 for l5 l1 and l2 for l6 l1 and l2 for l8 L1 Adapted Practice Name Class Practice 3-3 L3 Date 16. a. Which of the numbered angles at the right are exterior angles? l5, l6, l8 b. Name the remote interior angles for each. c. How are exterior angles 6 and 8 related? They are O vert. '. 17. a. How many exterior angles at the right are at each vertex of the triangle? 2 b. How many exterior angles does a triangle have in all? 6 Example 3 L4 Enrichment Practice Parallel Lines and the Triangle Angle-Sum Theorem Find the value of each variable. 1. 2. 3 y 65 x 30 60 n 39 75 68 4. 5. 93 6. 61 8. 10 55 a b c 44 25 53 y w 11. 1 12. 2 32 © Pearson Education, Inc. All rights reserved. 140 14. 2 69.7 18. 62 v t 4 60 115.5 3 15. 2 116 1 1 2 45 13 3 4 38 31 20. 128.5 5 120 46 126.8 3 19. 123 70 72 86 ml3 92; ml4 88 56 63 16. The sides of a triangle are 10 cm, 8 cm, and 10 cm. Classify the triangle. 17. The angles of a triangle are 44°, 110°, and 26°. Classify the triangle. 150 Use a protractor and a centimeter ruler to measure the angles and the sides of each triangle. Classify each triangle by its angles and sides. 18. 19. Chapter 3 Parallel and Perpendicular Lines 20. 12. 150 4 5 3 8 6 7 x 2 Algebra Find each missing angle measure. 9. 28 z x p x Find the measure of each numbered angle. 13. 2 46 79 10. 1 m 36 7. 15. scalene acute L3 13. 14. 15. 47 (page 149) a B Connection to Music 21. Music The lid of a grand piano is held open by a prop stick whose length can vary, depending upon the effect desired. The longest prop stick makes angles as shown. What are the values of x and y? x 147, y 33 22. A short prop stick makes the angles shown below. What are the values of a and b? a 162, b 18 Example 4 Exercise 21 Ask students what y x other instruments can modify volume by blocking the escape of sound waves. For example, trumpets and trombones have “mutes” that block sound when inserted into the bells, and French horn players place their fists in the bells of their instruments to reduce the volume. 57 Exercises 23–26 Done together as a class, these exercises provide a good opportunity to review the cumulative knowledge of students. Ask students to justify their solutions step by step for the rest of the class, giving the names or descriptions of the theorems and postulates they use to find the angle measures. 72 b Apply Your Skills x 2 Algebra Find the values of the variables and then the measures of the angles. Classify each triangle by its angles. Note that some figures have more than one triangle. x 7; 55, (8x 1) 35, 90; right 23. 24. (2x 4) (2x 9) (4x 7) x 37; 37, 65, 78; acute Exercises 25, 26 Remind students to begin each problem by asking: How many triangles are in the diagram? x Exercise 30 Students can work 25. GPS B 26. a 67, b 58, c 125, d 23, e 90; kFGH: 58, 67, 55; acute; kFEH: 125, 32, 23; obtuse; 54 A kEFG: 67, 23, 90; right y x z D x 38, y 36, 26. See left. z 90; kABD: E d 36, 90, 54; right; kBCD: 90, 52, 38; right; kABC: 74, 52 52, 54; acute C e 32 c 55 H F b a G 27. Reasoning What is the measure of each angle of an equiangular triangle? Explain. 60; 180 3 60 28. Writing Is every equilateral triangle isosceles? Is every isosceles triangle equilateral? Explain. See margin. 29. Visualization The diagram shows a triangle on a 3-by-3 geoboard. How many triangles with different shapes can be made on this geoboard? Classify each triangle by its sides and angles. See margin. 30. Multiple Choice The measure of one angle of a triangle is 115. The other two angles are congruent. What is the measure of each? A 32.5 65 57.5 115 Problem Solving Hint x 2 31. Algebra A right triangle has acute angles whose measures are in the ratio 1 : 2. In Exercise 31, use x and 2x for the angle measures. In Exercise 32, use 2x, 3x, and 4x. x 32. a. Algebra The ratio of the angle measures in #BCR is 2 : 3 : 4. Find the backward by multiplying each answer choice by 2 and adding their result to 115. Alternatively, they can subtract 115 from 180, and divide their result by 2. Exercise 32 To help students understand why 2x, 3x, and 4x are good representations of the angle measures, have students choose values for x and calculate the ratio of the angle measures. Diversity Exercise 37 The textile art of some countries features patterns of triangles. Students may be able to describe, show photographs of, or bring in examples of such textiles. Find the measures of these angles. 30 and 60 2 angle measures. 40, 60, 80 b. What type of triangle is #BCR? acute Lesson 3-4 Parallel Lines and the Triangle Angle-Sum Theorem 28. Yes, an equilateral k is isosc. because if three sides of a k are O, then two sides are O. No, the third side of an isosc. k does not need to be O to the other two. 29. eight 151 Acute isosceles Right isosceles Right scalene Obtuse scalene 151 4. Assess & Reteach GO PowerPoint Homework Help Visit: PHSchool.com Web Code: aue-0304 Lesson Quiz 1. A triangle with a 90° angle has sides that are 3 cm, 4 cm, and 5 cm long. Classify the triangle by its sides and angles. scalene right triangle Use the diagram for Exercises 2–6. 33. Draw any triangle. Label it #ABC. Extend both sides of the triangle to form two exterior angles at vertex A. Use the two exterior angles to explain why it does not matter which side of a triangle is extended to form an exterior angle. nline 33. Check students’ work. Answers may vary. Sample: The two exterior ' formed at vertex A are vertical ' and thus have the same measure. Proof 34. Prove the following theorem. 34-35. See margin. The acute angles of a right triangle are complementary. B Given: #ABC with right angle C Prove: &A and &B are complementary. Proof A C 35. Prove the Triangle Exterior Angle Theorem. 2 Given: &1 is an exterior angle of the triangle. Prove: m&1 = m&2 + m&3 1 4 3 3 1 2 4 36. Reasoning Two angles of a triangle measure 64 and 48. Find the measure of the largest exterior angle. Explain. See margin. 5 Real-World 2. Find m&3 if m&2 = 70 and m&4 = 42. 68 3. Find m&5 if m&2 = 76 and m&3 = 90. 166 Connection Patricia Watson Tsinnie often uses isosceles triangles in her rug designs. 37. Open-Ended Study the design in the Navajo weaving below. Make a design of your own that makes repeated use of isosceles triangles. Check students’ work. 4. Find x if m&1 = 4x, m&3 = 2x + 28, and m&4 = 32. 30 5. Find x if m&2 = 10x, m&3 = 5x + 40, and m&4 = 3x - 4. 8 6. Find m&3 if m&1 = 125 and m&5 = 160. 105 Alternative Assessment Have students draw and label a triangle with an exterior angle at each vertex. They should measure and label each angle, classify the triangle, and then explain how the measurements illustrate the Triangle Angle-Sum Theorem and the Triangle Exterior Angle Theorem. 38. The measures of the angles of #RST are 5!x, 7 !x, and 8!x. a. Find the value of x. 81 b. Give the measure of each angle. 45, 63, 72 c. What type of triangle is #RST? acute C Challenge Test Prep Resources CD bisects &ACB. Find m&DBF. 115 For additional practice with a variety of test item formats: • Standardized Test Prep, p. 193 • Test-Taking Strategies, p.188 • Test-Taking Strategies with Transparencies 41. What can you conclude about the bisector of an exterior angle of a triangle if the remote interior angles are congruent? Justify your response. See margin. 152 34. By the definition of right angle, mlC ≠ 90. By the Triangle Angle-Sum Theorem, mlA ± mlB ± mlC ≠ 180. 152 39. Reasoning Sketch a triangle and two exterior angles that have a side of the triangle in common. For what type of triangle, if any, is each statement true? Justify each answer. See back of book. a. The bisectors of the two exterior angles are parallel. b. The bisectors of the two exterior angles are perpendicular. c. The bisectors of the two exterior angles and the common side of the given triangle form an isosceles triangle. A 40. In the figure at the right, CD ' AB and D F B (3x 2) C (5x 20) Chapter 3 Parallel and Perpendicular Lines Subtracting 90 from each side gives mlA ± mlB ≠ 90, so lA and lB are complementary by the definition of comp. angles. 35. ml1 ± ml4 ≠ 180 by the l Add. Postulate. ml2 ± ml3 ± ml4 ≠ 180 by the k l-Sum Theorem. ml1 ± ml4 ± ml2 ± ml3 ± ml4 by the Trans. Property of Equality. ml1 ± ml2 ± ml3 by the Subtr. Property of Equality. Test Prep Multiple Choice Use the diagram at the right for Exercises 42–44. Use this Checkpoint Quiz to check students’ understanding of the skills and concepts of Lessons 3-1 through 3-4. M 42. m&M = 25 and m&L = 43. What is m&JKM? G F. 18 G. 68 H. 117 J. 162 Resources 43. m&M = 4x, m&L = 5x, and m&MKL = 6x. What is m&JKM? B A. 72 B. 108 C. 120 D. 132 J K L Grab & Go • Checkpoint Quiz 1 44. m&JKM = 15x - 48, m&L = 5x + 12, and m&M = 40. What is m&MKL? H F. 9 G. 57 H. 78 J. 97 Mixed Review GO for Help Lesson 3-3 Use the diagram at the right for Exercises 45–46. 1 a 45. If &1 and &2 are supplementary, what can you conclude about lines a and c? Justify your answer. b 46. If a 6 c, what can you conclude about lines a and b? Justify your answer. a n b; two lines n to the same line are n to each other. Lesson 1-6 x 2 47. Algebra In the figure at the right, A m&AOB = 3x + 20, m&BOC = x + 32, and m&AOC = 80. Find the value of x. 7 c 45. a n c by the Conv. of the Same-Side Ext. ' Thm. Lesson 1-1 O Draw the next figure in each sequence. 48. B C 49. Checkpoint Quiz 1 7. Converse of Alternate Exterior Angles Theorem 2 Lessons 3-1 through 3-4 Use the diagram at the right for Exercises 1–9. State the theorem or postulate that justifies each statement. Corr. ' Postulate Conv. of Corr. ' Post. a b c 1. &1 > &3 2. If &5 > &9, then d 6 e. 3. Same-Side Int. ' Thm. 1 2 3 3. m&1 + m&2 = 180 4. If &4 > &7, then d 6 e. d 4. Conv. of the Alt. Int. ' Thm. 4 5 6 5. &1 > &4 6. &7 > &9 Vertical ' Theorem Alt. Int. ' Thm. 7 8 7. If &3 > &9, then d 6 e. 8. &4 > &5 e 9 Corr. ' Postulate 9. If e ' b, then e ' c. If a line is ' to 1 of 2 parallel lines, it is ' to both. (2x 23) 10. Find the measures of the angles of each triangle. Classify each triangle by its angles. w x y 38, 55, 87; acute 38 55, 26, 99; obtuse (4w 5) lesson quiz, PHSchool.com, Web Code: aua-0304 36. 132; since the l is 68, the largest l is 180 – 48 ≠ 132. 41. Answers may vary. Sample: The measure of the ext. l is ≠ to the sum of the measures of the Lesson 3-4 The Triangle Angle-Sum Theorem two remote int. '. Since these ' are O, the ' formed by the bisector of the ext. l are O to each of them. Therefore, the bisector is n to the included side of the 153 remote ' by the Conv. of the Alt. Int. ' Thm. 153