Parallel Lines and the Triangle Angle-Sum Theorem. Classify each angle as acute, right, or obtuse

Transcription

1 - What You ll Learn To classify triangles and find the measures of their angles To use exterior angles of triangles... nd Why To find the reclining angle of a lounge chair, as in Example Parallel Lines and the Triangle ngle-sum Theorem heck Skills You ll Need GO for Help lassify each angle as acute, right, or obtuse.... right acute acute Solve each equation x = x + 05 = x + 58 = x = New Vocabulary acute triangle right triangle obtuse triangle Lesson -6 equiangular triangle equilateral triangle isosceles triangle scalene triangle exterior angle of a polygon remote interior angles -. Plan Objectives To classify triangles and find the measures of their angles To use exterior angles of triangles Examples pplying the Triangle ngle-sum Theorem lassifying a Triangle Using the Exterior ngle Theorem Real-World onnection Math ackground Finding ngle Measures in Triangles The diagrams at the top of the ctivity Lab on page 6 suggest the Triangle-ngle Sum Theorem. Key oncepts Theorem - Triangle ngle-sum Theorem The sum of the measures of the angles of a triangle is 80. m& + m& + m& = 80 The following proof of Theorem - relies on the idea that through a point not on a given line you can draw a line parallel to the given line. When alterations of Euclid s Parallel Postulate lead to different geometries, the Triangle ngle- Sum Theorem appears strikingly different. In a hyperbolic geometry, the sum of a triangle s angle measures is less than 80; in an elliptic geometry, the sum is greater than 80. More Math ackground: p. Lesson Planning and See p. E for a list of the resources that support this lesson. Proof Proof of Theorem - Given: # P Prove: m& + m& + m& = 80 Proof: ) y the Protractor Postulate, you can draw P so that m& = m&. Then, & and & are * ) congruent alternate interior angles, so P 6. & and & are also alternate interior angles, so by the lternate Interior ngles Theorem, m& = m&. y substitution, m& + m& + m& = m& + m& + m&, which is equal to 80 by the ngle ddition Postulate. Lesson - Parallel Lines and the Triangle ngle-sum Theorem 7 ell Ringer Practice heck Skills You ll Need For intervention, direct students to: Finding ngle Measures Lesson -6; Example Extra Skills, Word Problems, Proof Practice, h. Solving Linear Equations lgebra Review, page 0 Special Needs L 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 elow Level L Using geometry software to draw, measure, and manipulate the seven triangles on page 8 will help students discover results such as the Isosceles Triangle Theorem. learning style: visual 7

2 . Teach Guided Instruction Teaching Tip 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 ngle-sum Theorem as: & S EXMPLE lternative Method efore finding any values, ask: How many triangles are in the diagram? Use the question to highlight the alternate method of first finding the value of z using GFH. Quick heck nline EXMPLE pplying the Triangle ngle-sum Theorem lgebra Find the values of x and y. To find the value of x, use #GFJ x = 80 Triangle ngle-sum Theorem 0 + x = 80 Simplify. x = 76 Subtract 0 from each side. To find the value of y, look at &FJH. It is a straight angle. m&gjf + m&gjh = 80 x + y = 80 ngle ddition Postulate Substitute y = 80 Substitute 76 for x. y = 0 Subtract 76 from each side. Find the value of z in two different ways, each way using the Triangle ngle-sum Theorem. 55 In hapter, you classified an angle by its measure. You can also classify a triangle by its angles and sides. F 65 G 9 z J H EXMPLE Math Tip Students have not yet learned that a triangle is equilateral if and only if it is equiangular. s students see more triangles, encourage them to develop hypotheses about triangles, including what is impossible, such as an equilateral obtuse triangle. Visit: PHSchool.com Web ode: aue-0775 Equiangular cute Right Obtuse all angles congruent all angles acute one right angle one obtuse angle Equilateral Isosceles Scalene all sides congruent at least two sides congruent no sides congruent EXMPLE lassifying a Triangle dditional Examples lassify the triangle by its sides and its angles. In triangle, & is a right angle, and D #. Find the values of a, b, and c. 70 c a b D a = 70, b = 0, c = 0 lassify the triangle by its sides and its angles. 5 obtuse scalene Quick heck 8 hapter Parallel and Perpendicular Lines dvanced Learners L fter Example, help students discover the Triangle Exterior ngle Theorem by finding x and y as sums of angles in the triangles. 55, 65 9 t least two sides are congruent, so the triangle is isosceles. ll the angles are acute, so the triangle is acute. The triangle is an acute isosceles triangle. 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 English Language Learners ELL 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. 8 learning style: visual learning style: verbal

3 Using Exterior ngles of Triangles Vocabulary Tip Interior angle of a triangle means the same as angle of a triangle. n 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. The diagram at the right suggests a relationship between an exterior angle and its two remote interior angles. Theorem - states this relationship. You will prove this theorem in Exercise 5. Key oncepts Theorem - Triangle Exterior ngle Theorem The measure of each exterior angle of a triangle equals the sum of the measures of its two remote interior angles. m& = m& + m& Exterior angle Remote interior angles Guided Instruction Tactile Learners Encourage students to copy the triangle diagram shown immediately above the Triangle Exterior ngle 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 ngle Theorem. EXMPLE onnection to lgebra Suggest that students let x represent the measure of the angle to help them associate the equations with more familiar algebraic equations. EXMPLE Using the Exterior ngle Theorem lgebra Find each missing angle measure. a. b dditional Examples Find m&. 5 For: Triangle Theorems ctivity Use: Interactive Textbook, - D E D E D E D E 5 D E D E Quick heck Test-Taking Tip To check exterior angle-measure answers, remember that an exterior angle and the adjacent interior angle must be supplementary. m& = m& = 70 = 70 + m& = m& Two angles of a triangle measure 5. Find the measure of an exterior angle at each vertex. 5, 5, 90 EXMPLE Real-World onnection Multiple hoice The lounge chair has different settings that change the angles formed by its parts. Suppose m& is and m& is 8. Find m&, the angle formed by the back of the chair and the arm rest m& = m& + m& Exterior ngle Theorem m& = + 8 Substitute. m& = Simplify. The angle formed is a 8 angle. The correct choice is. 5 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. Daily Notetaking Guide - L Daily Notetaking Guide - dapted Instruction L losure Explain what is wrong with this diagram. Quick heck. a. b. Lesson - Parallel Lines and the Triangle ngle-sum Theorem D m&d must be greater than m&. c. Not possible; an equilateral k has all acute '.. Not possible; a right k will always have one longest side opp. the right l. 9

4 . Practice ssignment Guide -5, -5, 7-,, 7, 8 6-, 6,, 5, 6 hallenge 9- Test Prep - Mixed Review 5-8 Homework Quick heck To check students understanding of key skills and concepts, go over Exercises, 8, 5, 8,. Exercise 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. GPS Enrichment Guided Problem Solving Reteaching dapted Practice Practice Pearson Education, Inc. ll rights reserved. Name lass Date Practice - Find the value of each variable a b c z 8 Find the measure of each numbered angle m 6. The sides of a triangle are 0 cm, 8 cm, and 0 cm. lassify the triangle. 7. The angles of a triangle are, 0, and 6. lassify the triangle. Use a protractor and a centimeter ruler to measure the angles and the sides of each triangle. lassify each triangle by its angles and sides Parallel Lines and the Triangle ngle-sum Theorem 9 0 n w 68 p 8 5 v t L L 6 70 L L L 5 Quick heck EXERISES Example (page 8) Example (page 8) Example (page 9) x 50 hapter Parallel and Perpendicular Lines a. hange the setting on the lounge chair so that m& = and m& = 97. Find the new measure of &. 0 b. Explain how you can find m& without using the Exterior ngle Theorem. nswers may vary. Sample: Find the measure of the third l of the triangle. Subtract this from 80. For more exercises, see Extra Skill, Word Problem, and Proof Practice. Practice and Problem Solving GO Practice by Example for Help b. l and l for l5 l and l for l6 l and l for l8 Find ml lgebra Find the value of each variable x 70; y 0; z 0 80 x 80; y 80 z x Use a protractor and a centimeter ruler to measure the angles and the sides of each triangle. lassify each triangle by its angles and sides 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. 0. acute equilateral 0 5. See margin. equilateral right. obtuse scalene pp obtuse isosceles. isosceles right 5. scalene acute 6. 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. '. 7. a. How many exterior angles at the right are at each vertex of the triangle? b. How many exterior angles does a triangle have in all? 6 lgebra Find each missing angle measure acute, equiangular, equilateral c obtuse, isosceles 5 ml 9; ml

5 Example (page 9) pply Your Skills x 6. a 67, b 58, c 5, d, e 90; kfgh: 58, 67, 55; acute; kfeh: 5,, ; obtuse; kefg: 67,, 90; right Problem Solving Hint In Exercise, use x and x for the angle measures. In Exercise, use x, x, and x. x x. 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 7, y. short prop stick makes the angles shown below. What are the values of a and b? a 6, b 8 a b 7 lgebra Find the values of the variables and then the measures of the angles. lassify each triangle by its angles. Note that some figures have more than one triangle.. x 7; 55,. (8x ) 5, 90; right (x 7) 5 z 5 D e F d b c 55 H a 7. Reasoning What is the measure of each angle of an equiangular triangle? Explain. 60; Writing Is every equilateral triangle isosceles? Is every isosceles triangle equilateral? Explain. See margin. 9. Visualization The diagram shows a triangle on a -by- geoboard. How many triangles with different shapes can be made on this geoboard? lassify each triangle by its sides and angles. See margin. 0. Multiple hoice The measure of one angle of a triangle is 5. The other two angles are congruent. What is the measure of each? lgebra right triangle has acute angles whose measures are in the ratio :. Find the measures of these angles. 0 and 60. a. lgebra The ratio of the angle measures in #R is : :. Find the angle measures. 0, 60, 80 b. What type of triangle is #R? acute E (x ) x 8, y 6, See left. z 90; kd: GPS 6, 90, 5; right; kd: 90, 5, 8; right; k: 7, 5, 5; acute (x 9) 57 x 7; 7, 65, 78; acute G onnection to Music Exercise sk students what 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. Exercises 6 Done together as a class, these exercises provide a good opportunity to review the cumulative knowledge of students. sk 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. Exercises 5, 6 Remind students to begin each problem by asking: How many triangles are in the diagram? Exercise 0 Students can work backward by multiplying each answer choice by and adding their result to 5. lternatively, they can subtract 5 from 80, and divide their result by. Exercise To help students understand why x, x, and x are good representations of the angle measures, have students choose values for x and calculate the ratio of the angle measures. Diversity Exercise 7 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. Lesson - Parallel Lines and the Triangle ngle-sum Theorem 5 8. 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. 9. eight cute isosceles Right isosceles Right scalene Obtuse scalene 5

6 . ssess & Reteach Lesson Quiz. triangle with a 90 angle has sides that are cm, cm, and 5 cm long. lassify the triangle by its sides and angles. scalene right triangle Use the diagram for Exercises 6.. Find m& if m& = 70 and m& =. 68. Find m&5 if m& = 76 and m& = Find x if m& = x, m& = x + 8, and m& = Find x if m& = 0x, m& = 5x + 0, and m& = x Find m& if m& = 5 and m&5 = GO Real-World nline Homework Help Visit: PHSchool.com Web ode: aue-00. heck students work. nswers may vary. Sample: The two exterior ' formed at vertex are vertical ' and thus have the same measure. onnection Patricia Watson Tsinnie often uses isosceles triangles in her rug designs.. Draw any triangle. Label it #. Extend both sides of the triangle to form two exterior angles at vertex. Use the two exterior angles to explain why it does not matter which side of a triangle is extended to form an exterior angle. Proof. Prove the following theorem. -5. See margin. The acute angles of a right triangle are complementary. Given: # with right angle Prove: & and & are complementary. Proof 5. Prove the Triangle Exterior ngle Theorem. Given: & is an exterior angle of the triangle. Prove: m& = m& + m& 6. Reasoning Two angles of a triangle measure 6 and 8. Find the measure of the largest exterior angle. Explain. See margin. 7. Open-Ended Study the design in the Navajo weaving below. Make a design of your own that makes repeated use of isosceles triangles. heck students work. lternative ssessment 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 ngle-sum Theorem and the Triangle Exterior ngle Theorem. Test Prep For additional practice with a variety of test item formats: Standardized Test Prep, p. 9 Test-Taking Strategies, p.88 Test-Taking Strategies with Transparencies hallenge 8. The measures of the angles of #RST are 5!x, 7!x, and 8!x. a. Find the value of x. 8 b. Give the measure of each angle. 5, 6, 7 c. What type of triangle is #RST? acute 9. 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. 0. In the figure at the right, D ' and D bisects &.Find m&df. 5. 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. F D (5x 0) (x ) 5 hapter Parallel and Perpendicular Lines. y the definition of right angle, ml 90. y the Triangle ngle-sum Theorem, ml ± ml ± ml Subtracting 90 from each side gives ml ± ml 90, so l and l are complementary by the definition of comp. angles. 5. ml ± ml 80 by the l dd. Postulate. ml ± ml ± ml 80 by the k l-sum Theorem. ml ± ml ± ml ± ml ± ml by the Trans. Property of Equality. ml ± ml ± ml by the Subtr. Property of Equality.

7 Test Prep Multiple hoice Use the diagram at the right for Exercises.. m&m = 5 and m&l =. What is m&jkm? G F. 8 G. 68 H. 7 J. 6. m&m = x, m&l = 5x, and m&mkl = 6x. What is m&jkm? D.. m&jkm = 5x - 8, m&l = 5x +, and m&m = 0. What is m&mkl? H F. 9 G. 57 H. 78 J. 97 J K M L Use this heckpoint Quiz to check students understanding of the skills and concepts of Lessons - through -. Grab & Go heckpoint Quiz Mixed Review GO for Help Lesson - 5. a n c by the onv. of the Same-Side Ext. ' Thm. Lesson -6 Lesson - x Use the diagram at the right for Exercises If & and & are supplementary, what can you conclude about lines a and c? Justify your answer. 6. 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. 7. lgebra In the figure at the right, m&o = x + 0, m&o = x +, and m&o = 80. Find the value of x. 7 Draw the next figure in each sequence. O a b c heckpoint Quiz Lessons - through - 7. onverse of lternate Exterior ngles Theorem Use the diagram at the right for Exercises 9. State the theorem or postulate that justifies each statement. orr. ' Postulate onv. of orr. ' Post.. & > &. If &5 > &9, then d 6 e. a b c. Same-Side Int. ' Thm.. m& + m& = 80. If & > &7, then d 6 e.. onv. of the lt. Int. ' Thm. d & > & 6. &7 > &9 Vertical ' Theorem lt. Int. ' Thm. 7. If & > &9, then d 6 e. 8. & > &5 7 8 e orr. ' Postulate 9 9. If e ' b, then e ' c. If a line is ' to of parallel lines, it is ' to both. 0. Find the measures of the angles (x ) of each triangle. lassify each triangle by its angles. w 8, 55, 87; acute 8 55, 6, 99; obtuse (w 5) lesson quiz, PHSchool.com, Web ode: aua-00 Lesson - The Triangle ngle-sum Theorem 5 6. ; since the l is 68, the largest l is nswers may vary. Sample: The measure of the ext. l is to the sum of the measures of the 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 remote ' by the onv. of the lt. Int. ' Thm. 5