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3-1 Lines and Angles Vocabulary Review Write T for true or F for false. T 1. You can name a plane by a capital letter, such as A. F 2. A plane contains a finite number of lines. T 3. Two points lying on the same plane are coplanar. T 4. If two distinct planes intersect, then they intersect in exactly one line. Vocabulary Builder PA The symbol for parallel is . ruh lel Definition: Parallel lines lie in the same plane but never intersect, no matter how far they extend. Use Your Vocabulary 5. Circle the segment(s) that are parallel to the x-axis. AB BC CD AD 6. Circle the segment(s) that are parallel to the y-axis. AB BC A CD Ľ5 Ľ4 Ľ3 Ľ2 Ľ1 O AD D 7. Circle the polygon(s) that have two pairs of parallel sides. rectangle parallelogram square trapezoid Complete each statement below with line or segment. 8. A 9 consist of two endpoints and all the points between them. segment 9. A 9 is made up of an infinite number of points. line Chapter 3 58 3 2 1 Ľ2 Ľ3 y B x 1 2 3 4 5 C Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved. parallel (noun) Key Concept Parallel and Skew Parallel lines are coplanar lines that do not intersect. C D Skew lines are noncoplanar; they are not parallel and do not intersect. A Parallel planes are planes that do not intersect. H 10. Write each word, phrase, or symbol in the correct oval. noncoplanar coplanar intersect * ) do not intersect * ) * ) AE and CG Use arrows to show X X X X AE I BF and AD I BC. CB and AE Parallel Skew coplanar noncoplanar do not intersect do not intersect AE and CG CB and AE * ) * ) * ) G F E * ) B * ) Problem 1 Identifying Nonintersecting Lines and Planes Got It? Use the figure at the right. Which segments are parallel to AD? C B 11. In plane ADHE, EH is parallel to AD. A Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved. 12. In plane ADBC, BC is parallel to AD . 13. In plane ADGF, FG is parallel to AD . Got It? Reasoning Explain why FE and CD are not skew. D F E G H 14. Cross out the words or phrases below that do NOT describe skew lines. coplanar do not intersect parallel intersect noncoplanar not parallel 15. Circle the correct statement below. Segments and rays can be skew if they lie in skew lines. Segments and rays are never skew. 16. Underline the correct words to complete the sentence. FE and CD are in a plane that slopes from the bottom / top left edge to the bottom / top right edge of the figure. 17. Why are FE and CD NOT skew? Answers may vary. Sample: They are not skew segments because _______________________________________________________________________ they are part of the plane CDEF. Therefore, they are coplanar. _______________________________________________________________________ 59 Lesson 3-1 Key Concept Angle Pairs Formed by Transversals t Alternate interior angles are nonadjacent interior angles that lie on opposite sides of the transversal. Exterior 1 2 4 3 Same-side interior angles are interior angles that lie on the same side of the transversal. Corresponding angles lie on the same side of a transversal t and in corresponding positions. Interior 5 6 8 7 m Alternate exterior angles are nonadjacent exterior angles that lie on opposite sides of the transversal. Exterior Use the diagram above. Draw a line from each angle pair in Column A to its description in Column B. Column A Column B 18. /4 and /6 alternate exterior angles 19. /3 and /6 same-side interior angles 20. /2 and /6 alternate interior angles 21. /2 and /8 corresponding angles Identifying an Angle Pair Got It? What are three pairs of corresponding angles in the diagram at m the right? 1 2 8 7 Underline the correct word(s) or letter(s) to complete each sentence. 22. The transversal is line m / n / r . n 3 4 6 5 r 23. Corresponding angles are on the same side / different sides of the transversal. 24. Name three pairs of corresponding angles. Answers may vary. Accept any three of / 1 and / 3 / 2 and / 4 / 8 and / 6 / 7 and / 5 Problem 3 Classifying an Angle Pair Got It? Are angles 1 and 3 alternate interior angles, same-side interior 1 2 angles, corresponding angles, or alternate exterior angles? 25. Are /1 and /3 on the same side of the transversal? Yes / No 26. Cross out the angle types that do NOT describe /1 and /3. alternate exterior alternate interior corresponding 27. /1 and /3 are 9 angles. Chapter 3 corresponding 60 same-side interior 4 3 Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved. Problem 2 Lesson Check • Do you know HOW? Name one pair each of the segments or planes. Answers may vary. Samples are given. F 28. parallel segments 29. skew segments 30. parallel planes B E EFGH HD and BC ABCD 6 AB 6 EF A Name one pair each of the angles. 31. alternate interior 32. same-side interior /8 and / 6 8 /8 and / 3 33. corresponding 5 34. alternate exterior /1 and / 3 3 4 6 H G C D 1 7 2 /7 and / 5 Lesson Check • Do you UNDERSTAND? Error Analysis Carly and Juan examine the figure at the right. Carly says AB 6 HG. Juan says AB and HG are skew. Who is correct? Explain. D A B G Write T for true or F for false. Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved. C H F T 35. Parallel segments are coplanar. F 36. There are only six planes in a cube. F 37. No plane contains AB and HG . E 38. Who is correct? Explain. Carly; Sample explanation: The segments are parallel since they do _______________________________________________________________________ not intersect and are coplanar (plane ABGH contains AB and HG ). _______________________________________________________________________ Math Success Check off the vocabulary words that you understand. angle parallel skew transversal Rate how well you can classify angle pairs. Need to review 0 2 4 6 8 Now I get it! 10 61 Lesson 3-1 Properties of Parallel Lines 3-2 Vocabulary Review > 1. Circle the symbol for congruent. 5 6 Identify each angle below as acute, obtuse, or right. 2. 3. 4. 72í 125í obtuse right acute Vocabulary Builder E interior (noun) in TEER ee ur m interior Related Words: inside (noun), exterior (noun, antonym) Definition: The interior of a pair of lines is the region between the two lines. Example: A painter uses interior paint for the inside of a house. Use Your Vocabulary Use the diagram at the right for Exercises 5 and 6. Underline the correct point to complete each sentence. B A 5. The interior of the circle contains point A / B / C . C 6. The interior of the angle contains point A / B / C . 7. Underline the correct word to complete the sentence. The endpoint of an angle is called its ray / vertex . A 8. Write two other names for /ABC in the diagram at the right. l1 Chapter 3 lB 62 B 1 C Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved. Main Idea: The interior is the inside of a figure. Postulate 3-1, Theorems 3-1, 3-2, 3-3 Then... corresponding angles are congruent. If... Theorem 3-1 Alternate Interior Angles Theorem alternate interior angles are congruent. a transversal Theorem 3-2 Corresponding Angles Theorem intersects two parallel lines, same-side interior angles are supplementary. alternate exterior angles are congruent. Postulate 3-1 Same-Side Interior Angles Postulate Theorem 3-3 Alternate Exterior Angles Theorem Use the graphic organizer and the diagram to find each congruent angle. 9. Theorem 3-2 10. Theorem 3-1 l1 /3 > l7 /3 > 4 5 3 6 2 7 1 8 11. Theorem 3-3 l5 /1 > r s HSM11_GEMC_0302_T93307 Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved. HSM11_GEMC_0302_T93303 Problem 1 Identifying Congruent Angles Got It? Reasoning Can you always find the measure of all 8 angles when two parallel lines are cut by a transversal? Explain. 1 Yes, because m/1 5 55 by the Vertical Angles Theorem. m/5 5 55 by the Corresponding Angles Postulate because /1 and /5 are corresponding angles. 5 2 4 55 6 8 7 12. Write a reason for each statement. Answers may vary. Sample: m/7 5 55 Corresponding Angles Postulate m/5 5 m/7 Vertical Angles Theorem m/5 5 55 Transitive Property of Equality m/2 5 125 Same-Side Interior Angles Postulate m/4 5 125 Vertical Angles Theorem m/6 and m/8 5 125 Corresponding Angles Postulate 63 hsm11gmse_0302_t00237.ai Lesson 3-2 Problem 2 Proving an Angle Relationship Got It? Given: a 6 b a 1 Prove: /1 > /7 13. Use the reasons at the right to write each step of the proof. Statements b Reasons 5 8 7 anb 1) 1) Given l1 > l5 2) 2) If lines are 6, then corresp. angles are >. ml1 5 ml5 3) hsm11gmse_0302_t00240.ai 3) Congruent angles have equal measure. l5 > l7 4) 4) Vertical angles are congruent. ml5 5 ml7 5) 5) Congruent angles have equal measure. ml1 5 ml7 6) 6) Transitive Property of > l1 > l7 7) 7) Angles with equal measure are >. Problem 3 Finding Measures of Angles 14. There are two sets of parallel lines. Each parallel line also acts as a 9. transversal 15. The steps to find m/1 are given below. Justify each step. Statements m p q 2 1 8 4 6 5 3 105 7 Reasons 1)/1 > /4 1) Alternate Interior Angles Theorem 2)m/1 5 m/4 the same measure. 2) Congruent angles have hsm11gmse_0302_t00242.ai 3)/4 and /6 are supplementary. 3) Linear Pair Postulate 4)m/4 1 m/6 5 180 4) Definition of supplementary angles 5)m/1 1 m/6 5 180 5) Transitive Property of Equality 6)m/5 5 105 6) Corresponding angles have the same measure. 7)m/6 5 105 7) Alternate interior angles have the same measure. 8)m/1 1 105 5 180 8) Substitute into Statement 5. 9)m/1 5 75 9) Subtraction Property of Equality Chapter 3 64 Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved. Got It? Find the measure of l1. Justify your answer. Problem 4 Using Algebra to Find an Angle Measure Got It? In the figure at the right, what are the values of x and y? 2x 3y 16. The bases of a trapezoid are parallel / perpendicular . 17. Use the Same-Side Interior Angles Postulate to complete each statement. (y 20) (x 12) y 1 20 5 180 3y 1 x 2 12 5 180 2x 1 18. Solve each equation. 2x 1 (x 2 12) 5 180, 3x 2 12 5 180 3x 5 192 x 5 64 3y 1 (y 1 20) 5 180 4y 1 20 5 180 4y 5 160 y 5 40 hsm11gmse_0302_t00244.ai Lesson Check • Do you UNDERSTAND? In the diagram at the right, l1 and l8 are supplementary. What is a good name for this pair of angles? Explain. a 19. Circle the best name for lines a and b. b parallel perpendicular skew transversals 1 5 8 7 20. Circle the best name from the list below for /1 and /8. alternate congruent corresponding same-side Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved. 21. Circle the best name from the list below for /1 and /8. exterior interior hsm11gmse_0302_t00240.ai 22. Use your answers to Exercises 20 and 21 to write a name for /1 and /8. same-side exterior angles ________________________________________________________________________ Math Success Check off the vocabulary words that you understand. alternate interior angles alternate exterior angles Rate how well you can prove angle relationships. Need to review 0 2 4 6 8 Now I get it! 10 65 Lesson 3-2 3-3 Proving Lines Parallel Vocabulary Review Write the converse of each statement. 1. Statement: If you are cold, then you wear a sweater. Converse: If 9, then 9. If you wear a sweater , then you are cold . 2. Statement: If an angle is a right angle, then it measures 90°. Converse: If an angle measures 90°, then it is a right angle. 3. The converse of a true statement is always / sometimes / never true . Vocabulary Builder E m Related Words: exterior (noun), external, interior (antonym) exterior Definition: Exterior means on the outside or in an outer region. Example: Two lines crossed by a transversal form four exterior angles. Use Your Vocabulary Underline the correct word to complete each sentence. 4. To paint the outside of your house, buy interior / exterior paint. 5. The protective cover prevents the interior / exterior of the book from being damaged. 6. In the diagram at the right, angles 1 and 7 are alternate interior / exterior angles. 7. In the diagram at the right, angles 4 and 5 are same-side interior / exterior angles. Underline the hypothesis and circle the conclusion in the following statements. 8. If the lines do not intersect, then they are parallel lines. 9. If the angle measures 180˚, then it is a straight angle. Chapter 3 66 1 2 4 3 5 6 8 7 E m Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved. exterior exterior (adjective) ek STEER ee ur Theorems 3-2 and 3-4 Corresponding Angles Theorem and Its Converse Theorem 3-2 Corresponding Angles Theorem If a transversal intersects two parallel lines, then corresponding angles are congruent. 10. Complete the statement of Theorem 3-2. Theorem 3-4 Converse of the Corresponding Angles Theorem If two lines on a transversal form corresponding angles that are congruent, then the lines are 9. parallel 11. Use the diagram below. Place appropriate marking(s) to show that /1 and /2 are congruent. r s 1 2 t 12. Circle the diagram that models Theorem 3-4. 1 HSM11_GEMC_0303_T93477 ℓ 2 1 m 2 ℓ m Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved. Theorems 3-5, 3-6, and 3-7 HSM11_GEMC_0303_T93311 HSM11_GEMC_0303_T93310 Theorem 3-5 Converse of the Alternate Interior Angles Theorem If two lines and a transversal form alternate interior angles that are congruent, then the two lines are parallel. Theorem 3-6 Converse of the Same-Side Interior Angles Theorem If two lines and a transversal form same-side interior angles that are supplementary, then the two lines are parallel. Theorem 3-7 Converse of the Alternate Exterior Angles Theorem If two lines and a transversal form alternate exterior angles that are congruent, then the two lines are parallel. 13. Use the diagram at the right to complete each example. Theorem 3-5 l8 , If /4 > then b6c. Theorem 3-6 Theorem 3-7 l8 If /3 and are supplementary, then b6c. l7 , If /1 > then b6c. 6 7 5 8 4 3 1 2 b c HSM11_GEMC_0303_T93308 67 Lesson 3-3 Problem 1 Identifying Parallel Lines Got It? Which lines are parallel if l6 O l7? Justify your answer. 14. Underline the correct word(s) to complete each sentence. /6 > /7 is given / to prove . a 3 b 4 5 m 6 1 8 2 7 /6 and /7 are alternate / same-side angles. /6 and /7 are corresponding / exterior / interior angles. I can use Theorem 3-2 / Theorem 3-4 to prove the lines parallel. Using /6 > /7, lines a and b / and m are parallel and the transversal is a/ b / / m . hsm11gmse_0303_t00246.ai Problem 2 Writing a Flow Proof of Theorem 3-6 Got It? Given that l1 O l7. Prove that l3 O l5 using a flow proof. 15. Use the diagram at the right to complete the flow proof below. ∠1 ≅ ∠7 1 m 5 6 3 7 ∠7 ≅ ∠5 ∠3 ≅ ∠7 ∠3 ≅ ∠5 ∠1 ≅ ∠3 Transitive Prop. Vertical angles Transitive Prop. hsm11gmse_0303_t00252 Vertical angles of ≅. are ≅. of ≅. are ≅. Problem 3 Determining Whether Lines Are Parallel Got It? Given that l1 O l2, you can use the Converse of the Alternate Exterior Angles Theorem to prove that lines rHSM11_GEMC_0303_T93314 and s are parallel. What is another way to explain why r ns? Justify your answer. 16. Justify each step. 3 t 2 1 /1 > /2 Given /2 > /3 Vertical angles are congruent. /1 > /3 Transitive Property of Congruence 17. Angles 1 and 3 are alternate / corresponding . HSM11_GEMC_0303_T93315 18. What postulate or theorem can you now use to explain why r6s? Converse of the Corresponding Angles Theorem ________________________________________________________________________ Chapter 3 s r 68 Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved. Given Problem 4 Using Algebra Got It? What is the value of w for which c n d? c 55 Underline the correct word to complete each sentence. d (3w 2) 19. The marked angles are on opposite sides / the same side of the transversal. 20. By the Corresponding Angles Theorem, if c 6 d then corresponding angles are complementary / congruent / supplementary . 21. Use the theorem to solve for w. hsm11gmse_0303_t00255 3w 2 2 5 55 3w 5 57 w 5 19 Lesson Check • Do you UNDERSTAND? * ) * ) Error Analysis A classmate says that AB n DC based on the diagram at right. Explain your classmate's error. A 22. Circle the segments that are sides of /D and /C. Underline the transversal. AB BC DC DA D B 83 97 C Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved. 23. Explain your classmate’s error. Explanations may vary. Sample: My classmate identified the wrong pair of parallel sides. lADC and lBCD ________________________________________________________________________ hsm11gmse_0303_t003 are supplementary and same-side interior angles. The transversal is DC . ________________________________________________________________________ AD n BC by the Converse of the Same-Side Interior Angles Theorem. ________________________________________________________________________ Math Success Check off the vocabulary words that you understand. flow proof two-step proof parallel lines Rate how well you can prove that lines are parallel. Need to review 0 2 4 6 8 Now I get it! 10 69 Lesson 3-3 3-4 Parallel and Perpendicular Lines Vocabulary Review Complete each statement with always, sometimes or never. 1. A transversal 9 intersects at least two lines. always 2. A transversal 9 intersects two lines at more than two points. never 3. A transversal 9 intersects two parallel lines. sometimes 4. A transversal 9 forms angles with two other lines. always Vocabulary Builder Transitive transitive (adjective) TRAN si tiv If A B and B C then A C. Related Words: transition, transit, transitivity Main Idea: You use the Transitive Property in proofs when what you know implies a statement that, in turn, implies what you want to prove. Definition: Transitive describes the property where one element in relation to a second element and the second in relation to the third implies the first element is in relation to the third element. Use Your Vocabulary Complete each example of the Transitive Property. 5. If a . b 6. If Joe is younger than Ann 7. If you travel from and b . c, and Ann is younger than Station 2 to Station 3 then a S c . Sam, then and you travel from Joe is younger Station 3 to than Sam . Station 4 then you travel from Station 2 to Station 4. Chapter 3 70 , Theorem 3-8 Transitive Property of Parallel Lines and Theorem 3-9 8. Complete the table below. Theorem 3-8 Theorem 3-9 Transitive Property of Parallel Lines In a plane, if two line are perpendicular to the same line, then they are parallel to each other. If two lines are parallel to the same line, then they are parallel to each other. If a∙b m⊥t and b ∙ c n⊥t then a∙c m ∙ n HSM11_GEMC_0304_T13316 Problem 1 Solving a Problem With Parallel Lines Got It? Can you assemble the pieces at the right to form a picture frame with opposite sides parallel? Explain. 9. Circle the correct phrase to complete the sentence. 60 60 60 60 30˚ 30˚ To make the picture frame, you will glue 9. the same angle to the same angle two different angles together 10. The angles at each connecting end measure 60 8 and 30 8 . hsm11gmse_0304_t00260 90 8 . 11. When the pieces are glued together, each angle of the frame will measure 12. Complete the flow chart below with parallel or perpendicular. The left piece will be parallel The top and bottom to the right piece. Opposite sides are pieces will be parallel perpendicular to the side pieces. . The top piece will be parallel to the bottom piece. 13. Underline the correct words to complete the sentence. Yes / No , I can / cannot assemble the pieces to form a picture frame with opposite sides parallel. HSM11_GEMC_0304_T93317 71 Lesson 3-4 Theorem 3-10 Perpendicular Transversal Theorem n In a plane, if a line is perpendicular to one of two parallel lines, then it is also perpendicular to the other. 14. Place a right angle symbol in the diagram at the right to illustrate Theorem 3-10. ℓ m Use the information in each diagram to complete each statement. 15. a g n 16. t p HSM11_GEMC_0303_T93318 c g and a ' t , so g a 6 t . ' HSM11_GEMC_0304_T14343 c ' n and n 6 p , so c ' p . HSM11_GEMC_0304_T14344 Problem 2 Proving a Relationship Between Two lines Got It? Use the diagram at the right. In a plane, c ' b, b ' d, and d ' a. Can you conclude that anb? Explain. c 17. Circle the line(s) perpendicular to a. Underline the line(s) perpendicular to b. d a b c d a b 18. Lines that are perpendicular to the same line are parallel / perpendicular . 19. Can you conclude that a6b ? Explain. hsm11gmse_0304_t00263 Yes. Explanations may vary. Sample: Lines a and b are both ________________________________________________________________________ perpendicular to line d, so anb by Theorem 3-8. ________________________________________________________________________ Lesson Check • Do you know HOW? In one town, Avenue A is parallel to Avenue B. Avenue A is also perpendicular to Main Street. How are Avenue B and Main Street related? Explain. 20. Label the streets in the diagram A for Avenue A, B for Avenue B, and M for Main Street. M A B 21. Underline the correct word(s) to complete each sentence. The Perpendicular Transversal Theorem states that, in a plane, if a line is parallel / perpendicular to one of two parallel / perpendicular lines, then it is HSM11_GEMC_0304_T93918 also parallel / perpendicular to the other. Avenue B and Main Street are parallel / perpendicular streets. Chapter 3 72 Lesson Check • Do you UNDERSTAND? Which theorem or postulate from earlier in the chapter supports the conclusion in Theorem 3-9? In the Perpendicular Transversal Theorem? Explain. n Use the diagram at the right for Exercises 22 and 23. 22. Complete the conclusion to Theorem 3-9. ℓ In a plane, if two lines are perpendicular to the same line, then 9. m they are parallel to each other _______________________________________________________________ 23. Complete the statement of Theorem 3-4. If two lines and a transversal form 9 angles that are congruent, then the lines are parallel. corresponding HSM11_GEMC_0304_T93319 c Use the diagram at the right for Exercises 24 and 25. 24. Complete the conclusion to the Perpendicular Transversal Theorem. a In a plane, if a line is perpendicular to one of two parallel lines, then it is also 9. b perpendicular to the other. ________________________________________________________________________ 25. Explain how any congruent angle pairs formed by parallel lines support the conclusion to the Perpendicular Transversal Theorem. HSM11_GEMC_0304_T93320 Answers may vary. Sample: If both alternate interior angles are ________________________________________________________________________ right angles, then the line perpendicular to one parallel line is ________________________________________________________________________ perpendicular to the other parallel line. ________________________________________________________________________ ________________________________________________________________________ Math Success Check off the vocabulary words that you understand. parallel perpendicular Rate how well you can understand parallel and perpendicular lines. Need to review 0 2 4 6 8 Now I get it! 10 73 Lesson 3-4 Parallel Lines and Triangles 3-5 Vocabulary Review Identify the part of speech for the word alternate in each sentence below. 1. You vote for one winner and one alternate. noun 2. Your two friends alternate serves during tennis. verb 3. You and your sister babysit on alternate nights. adjective 4. Write the converse of the statement. Statement: If it is raining, raining then I need an umbrella. Converse: If I need an umbrella, then it is raining. tri- (prefix) try Related Word: triple Main Idea: Tri- is a prefix meaning three that is used to form compound words. Examples: triangle, tricycle, tripod Use Your Vocabulary Write T for true or F for false. T 5. A tripod is a stand that has three legs. F 6. A triangle is a polygon with three or more sides. F 7. A triatholon is a race with two events — swimming and bicycling. T 8. In order to triple an amount, multiply it by three. Chapter 3 74 Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved. Vocabulary Builder Postulate 3-2 Parallel Postulate P Through a point not on a line, there is one and only one line parallel to the given line. 1 line(s) through P parallel to line /. 9. You can draw Theorem 3-11 Triangle Angle-Sum Theorem hsm11gmse_0305_t00264 The sum of the measures of the angles of a triangle is 180. Find each angle measure. 10. C 11. M 45° 100° 30° B A L N 50 m/C 5 45 m/L 5 HSM11_GEMC_0305_T93322 HSM11_GEMC_0305_T93323 Problem 1 Using the Triangle Angle-Sum Theorem Got It? Use the diagram at the right. What is the value of z? Complete each statement. Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved. 59 12. m/A 5 43 A B 49 x y 59 D z C 43 1 49 5 92 13. m/ABC 5 180 14. m/A 1 m/ABC 1 m/C 5 hsm11gmse_0305_t00267 180 59 1 92 1 z 5 59 2 180 2 92 5 29 z 5 Check your result by solving for z another way. 15. Find m/BDA. 16. Then find m/BDC. mlA 1 mlABD 1 mlBDA 5 180 59 1 43 1 x 5 180 x 5 180 2 102 5 78 mlBDA 1 mlBDC 5 180 x 1 y 5 180 y 5 180 2 78 5 102 17. Use your answers to Exercises 15 and 16 to find the value of z. z 1 mlCBD 1 mlBDC 5 180 z 1 49 1 102 5 180 z 5 180 2 (49 1 102) 5 29 75 Lesson 3-5 Theorem 3-12 Triangle Exterior Angle Theorem 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. 2 The measure of each exterior angle of a triangle equals the sum of the measures of its two remote interior angles. 1 18. ml1 5 m/2 1 m/3 3 Circle the number of each exterior angle and draw a box around the number of each remote interior angle. 19. 20. 5 HSM11_GEMC_0305_T93324 3 6 4 1 2 HSM11_GEMC_0305_T93321 Exterior Angle Theorem Problem 2 Using the Triangle HSM11_GEMC_0305_T93330 Got It? Two angles of a triangle measure 53. What is the measure of an exterior angle at each vertex of the triangle? a Label the interior angles 538, 538, and a. Label the exterior angles adjacent to the 538 angles as x and y. Label the third exterior angle z. x 53° 53° y 22. Complete the flow chart. Triangle Angle-Sum 53 + 53 + a = 180 HSM11_GEMC_0305_T93325 a = 180 – 106 = 74 Exterior Angle x = a + 53 = 74 + 53 = 127 Chapter 3 Exterior Angle y = a + 53 = 74 + 53 = 127 76 HSM11_GEMC_0305_T93326 Exterior Angle z = 53 + 53 = 106 Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved. z 21. Use the diagram at the right. Problem 3 Applying the Triangle Theorems B 30í xí Got It? Reasoning Can you find mlA without using the A Triangle Exterior Angle Theorem? Explain. 80í 23. /ACB and /DCB are complementary / supplementary angles. 24. Find m/ACB. C D 180 2 80 5 100 25. Can you find m/A if you know two of the angle measures? Explain. Explanations may vary. Sample: Yes. Use the Triangle Angle-Sum Theorem. mlA 5 180 2 100 2 30 5 50 __________________________________________________________________________________ Lesson Check • Do you UNDERSTAND? 3 Explain how the Triangle Exterior Angle Theorem makes sense based on the Triangle Angle-Sum Theorem. 1 26. Use the triangle at the right to complete the diagram below. 2 4 mƋ1 à mƋ3 à mƋ2 â180 Triangle Angle-Sum Theorem mƋ1 à mƋ3 â mƋ4 Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved. Linear Pair Postulate à mƋ2 â180 mƋ4 27. Explain how the Triangle Exterior Angle Theorem makes sense based on the Triangle Angle-Sum Theorem. Answers may vary. Sample: Using the Triangle Angle-Sum Theorem,180 2 ml2 5 ml1 1 ml3. Since a linear pair is ______________________________________________________________________________________ supplementary, 180 2 ml2 5 ml4. Then by the Transitive Property, ml4 5 ml1 1 ml3. ______________________________________________________________________________________ Math Success Check off the vocabulary words that you understand. exterior angle remote interior angles Rate how well you can use the triangle theorems. Need to review 0 2 4 6 8 Now I get it! 10 77 Lesson 3-5 Constructing Parallel and Perpendicular Lines 3-6 Vocabulary Review Write T for true or F for false. T F 1. A rectangle has two pairs of parallel segments. 2. A rectangle has two pairs of perpendicular segments. Write alternate exterior, alternate interior, or corresponding to describe each angle pair. 4. 1 5. 5 3 2 4 alternate interior corresponding 6 alternate exterior Vocabulary Builder construction (noun) kun STRUCK shun Other Word Forms: construct (verb), constructive (adjective) Main Idea: Construction means how something is built or constructed. Math Usage: A construction is a geometric figure drawn using a straightedge and a compass. Use Your Vocabulary 6. Complete each statement with the correct form of the word construction. VERB You 9 sand castles at the beach. construct NOUN The 9 on the highway caused quite a traffic jam. construction ADJECTIVE The time you spent working on your homework was 9. constructive Chapter 3 78 Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved. 3. Problem 1 Constructing Parallel Lines Got It? Reasoning The diagram at the right shows the construction of line m through point N with line m parallel to line <. Why must lines < and m be parallel? m N 1 7. The diagram shows the construction of congruent H NHJ . angles 1 and J 8. Circle the description(s) of the angle pairs that were constructed. alternate interior congruent corresponding same-side interior vary. Sample: 9. Now explain why lines / and m must be parallel. Explanations mayhsm11gmse_0306_t00300.ai You constructed the transversal to form congruent ________________________________________________________________________ corresponding angles with lines < and m, so < n m by the Converse ________________________________________________________________________ of the Corresponding Angles Theorem. ________________________________________________________________________ Problem 2 Constructing a Special Quadrilateral *Got) It? * ) Draw a segment. Label its length m. Construct quadrilateral ABCD with AB n CD , so that AB 5 m and CD 5 2m. Underline the correct word or symbol to complete each sentence. 10. Construct parallel / perpendicular lines. * ) * ) * ) Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved. 11. Draw AB . Draw point D not on AB . Draw AD . The length of AB / AD is m. 12. At D, construct /TDZ perpendicular / congruent to /DAB so that /TDZ and * ) * ) /DAB are corresponding angles. Then DZ 6 AB . * ) 13. Now, you need a side of length 2m. Construct C on DZ so that DC 5 2m. Draw BC / BA . 14. Do the construction below. T D A Z m C B HSM11_GEMC_0306_T93919 79 Lesson 3-6 Problem 3 Perpendicular at a Point on a Line * ) * ) * ) * ) Got It? Use a straightedge to draw EF . Construct FG so that FG ' EF at point F. 15. Use the diagram at the right. Write each construction step. G Step 1 Draw a line. Label points E and F. __________________________________________ E F H Step 2 Make the compass opening EF. With the compass tip on point F, draw an arc that __________________________________________ intersects the line twice. Mark point H so __________________________________________ EF 5 FH. __________________________________________ HSM11_GEMC_0306_T93334 1 Step 3 Make the compass opening greater than 2 EH. With the compass tip on E, draw an arc above point F. __________________________________________________________________ __________________________________________________________________ Step 4 Without changing the compass setting, place the compass tip on point of intersection G. __________________________________________________________________ __________________________________________________________________ * ) Step 5 Draw FG . Postulate 3-3 Perpendicular Postulate Complete the statement of Postulate 3-3 below. 16. Through a point not on a line, there is one and only one line parallel / perpendicular to the given line. 17. Circle the diagram that models Postulate 3-3. P P P HSM11_GEMC_0306_T93920 HSM11_GEMC_0306_T93922 HSM11_GEMC_0306_T93921 Chapter 3 80 Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved. point H. Draw an arc that intersects the arc from Step 3. Label the __________________________________________________________________ Problem 4 Perpendicular From a Point to a Line * ) * ) * ) * ) * ) Got It? Draw a line CX and a point Z not on CX . Construct ZB so that ZB ' CX . Underline the correct word(s) to complete each sentence. 18. Open your compass to a size equal to / greater than the distance from Z to line /. 19. With the compass tip on point Z, draw an arc that intersects line / at one / two point(s). 20. Label the point(s) C and X / Z . 21. Place the compass point on C / Z and make an arc below line /. Z 22. With the same opening and the compass tip on C / X , draw an arc that intersects the arc you made in Exercise 21. Label the point of intersection B. C E X * ) * ) B 23. Draw ZB / CX . 24. Use line / and point Z at the right. Construct a line through point Z perpendicular to line /. Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved. Lesson Check • Do you UNDERSTAND? Suppose you use a wider compass setting in Exercise 18. Will you construct a different perpendicular line? Explain. 25. Explain why you will NOT construct a different perpendicular line. Explanations is one and only one line ___________________________________________________________________________ subtraction may vary. Sample: There multiplication perpendicular to a given line. ___________________________________________________________________________ Math Success Check off the vocabulary words that you understand. construction parallel perpendicular Rate how well you can construct parallel and perpendicular lines. Need to review 0 2 4 6 8 Now I get it! 10 81 Lesson 3-6 3-7 Equations of Lines in the Coordinate Plane Vocabulary Review Write T for true or F for false. T 1. An ordered pair describes the location of a point in a coordinate grid. F 2. An ordered pair can be written as (x-coordinate, y-coordinate) or (y-coordinate, x-coordinate). T 3. The ordered pair for the origin is (0, 0). Vocabulary Builder Slope â slope (noun, verb) slohp rise run Definition: The slope of a line m between two points (x1, y1) and (x2, y2) on a coordinate plane is the ratio of the vertical change (rise) to rise y 2y m 5 run 5 x2 2 x1 2 1 Use Your Vocabulary Complete each statement with the appropriate word from the list. Use each word only once. slope sloping sloped 4. The 9 of the hill made it difficult for bike riding. slope 5. The driveway 9 down to the garage. sloped 6. The 9 lawn led to the river. sloping Draw a line from each word in Column A to its corresponding part of speech in Column B. Column A Column B 7. linear ADJECTIVE 8. line NOUN Chapter 3 82 Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved. the horizontal change (run). Problem 1 Finding Slopes of Lines Got It? Use the graph at the right. What is the slope of line a? 9. Complete the table below to find the slope of line a. Think change in y-coordinates change in x-coordinates mâ . b y2 Ľy1 x2 Ľx1 (1, 7) (5, 7) 4 (2, 3) (1, 2) 2 x (4, 0) 6 O 2 8 (4, 2) 7 Ľ 3 Two points on line a are (2, 3) and (5, 7). a d 6 Write I know the slope is the ratio y c â 5 Ľ 2 Now I can simplify. 4 3 â Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved. Key Concept Forms of Linear Equations Definition Symbols The slope-intercept form of an equation of a nonvertical line is y 5 mx 1 b, where m is the slope and b is the y-intercept. y 5 mx 1 b c c slope y-intercept The point-slope form of an equation of a nonvertical line is y 2 y1 5 m(x 2 x1), where m is the slope and (x1, y1) is a point on the line. y 2 y1 5 m(x 2 c c y-coordinate slope x1) c x-coordinate Problem 2 Graphing Lines 5 Got It? Graph y 5 3x 2 4. y 4 3 10. In what form is the given equation written? 2 slope-intercept form _________________________________________ 11. Written as a fraction, the slope is 3 1 1 Ľ5 Ľ4 Ľ3 Ľ2 Ľ1 O Ľ1 . x 1 2 3 4 5 Ľ2 12. One point on the graph is ( 0 ,24). Ľ3 13. From that point, move 3 unit(s) up and 1 unit (s) to the right. Ľ4 Ľ5 14. Graph y 5 3x 2 4 on the coordinate plane. 83 Lesson 3-7 Writing Equations of Lines Problem 3 Got It? What is an equation of the line with slope 212 and y-intercept 2? 15. Complete the problem-solving model below. Know 1 slope m 5 22 y-intercept 5 2 Need Plan Write an equation of a line. Use y 5 mx 1 b , the slope-intercept form of a linear equation. 16. Now write the equation. y 5 212x 1 2 Problem 4 Using Two Points to Write an Equation Got It? You can use the two points given on the line at the right to y 17. The equation is found below. Write a justification for each step. Write in point-slope form. y 2 y1 5 m(x 2 x1) 6 Substitute. 6 Simplify. y 2 (21) 5 5(x 2 (22)) y 1 1 5 5(x 1 2) (ⴚ2, ⴚ1) ⫺3 Got It? Use the two equations for the line shown above. Rewrite the equations in slope-intercept form and compare them. What can you conclude? 18. Write each equation in slope-intercept form. 6 6 y 2 5 5 5(x 2 3) y 1 1 5 5(x 1 2) y 2 5 5 65x 2 18 5 25 y 5 65x 2 18 5 1 5 y 5 65x 1 75 y 1 1 5 65x 1 12 5 5 y 5 65x 1 12 5 25 y 5 65x 1 75 19. Underline the correct word(s) to complete each sentence. The equations are different / the same . Choosing (22,21) gives a different / the same equation as choosing (3, 5). 6 6 The equations y 2 5 5 5 (x 2 3) and y 1 1 5 5 (x 1 2) are / are not equivalent. Chapter 3 84 (3, 5) 4 O x 2 4 ⫺2 Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved. 6 show that the slope of the line is 5 . So one equation of the line is 6 y 2 5 5 5(x 2 3). What is an equation of the line if you use (22, 21) instead of (3, 5) in the point-slope form of the equation? Problem 5 Writing Equations of Horizontal and Vertical Lines Got It? What are the equations for the horizontal and vertical lines through (4, 23)? 5 Write T for true or F for false. y 4 T 20. Every point on a horizontal line through (4,23) has y-coordinate of 23. 3 2 1 F 21. The equation of a vertical line through (4,23) is y 5 23. T 22. The equation of a vertical line through (4,23) is x 5 4. Ľ5 Ľ4 Ľ3 Ľ2 Ľ1 O Ľ1 x 1 2 3 4 5 Ľ2 Ľ3 Ľ4 23. Graph the horizontal and vertical lines through (4,23) on the coordinate plane at the right. Ľ5 Lesson Check • Do you UNDERSTAND? Error Analysis A classmate found the slope of the line passing through (8,22) and (8, 10) as shown at the right. Describe your classmate’s error. Then find the correct slope of the line passing through the given points. 24. What is your classmate’s error? Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved. Answers may vary. Sample: Slope is rise over run, not run over rise. _________________________________________________________________ 88 10 (2) 0 m 12 m m0 25. Find the slope, m. 2 10 212 m 5 22 8 2 8 5 0 26. The run is 8 2 8 5 0 , so the slope is undefined . Math Success Check off the vocabulary words that you understand. slope slope-intercept form point-slope form Rate how well you can write and graph linear equations. Need to review 0 2 4 6 8 Now I get it! 10 85 Lesson 3-7 3-8 Slopes of Parallel and Perpendicular Lines Vocabulary Review y Use the graph at the right for Exercises 1–4. Write parallel or perpendicular to complete each sentence. b perpendicular 1. Line b is 9 to line a. x O 2. Line b is 9 to the x-axis. parallel 3. Line a is 9 to the y-axis. parallel 4. The x-axis is 9 to the y-axis. perpendicular a Write the converse, inverse, and contrapositive of the statement below. If a polygon is a triangle, then the sum of the measures of its angles is 180. the polygon is a triangle _______________________________________________________________________ 6. INVERSE If a polygon is not a triangle, then 9. the sum of the measures of its angles is not 180 _______________________________________________________________________ 7. CONTRAPOSITIVE If the sum of the measures of the angles of a polygon is not 180, then 9. the polygon is not a triangle _______________________________________________________________________ Vocabulary Builder The reciprocal of x is reciprocal (noun) rih SIP ruh kul Other Word Forms: reciprocate (verb) Definition: The reciprocal of a number is a number such that the product of the two numerator denominator numbers is 1. The reciprocal of denominator is numerator . Chapter 3 86 1 . x Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved. 5. CONVERSE If the sum of the measures of the angles of a polygon is 180, then 9. Use Your Vocabulary Complete each statement with reciprocal or reciprocate. Use each word only once. 8. VERB 9. NOUN After your friend helps you with your homework, you 9 by helping your friend with his chores. reciprocate 3 The 9 of 23 is 2 . reciprocal Key Concept Slopes of Parallel Lines • If two nonvertical lines are parallel, then their slopes are equal. • If the slopes of two distinct nonvertical lines are equal, then the lines are parallel. • Any two vertical lines or horizontal lines are parallel. Circle the correct statement in each exercise. 10. A vertical line is parallel to any other vertical line. 11. A vertical line is parallel to any horizontal line. Any two nonvertical lines have the same slope. Any two nonvertical lines that are parallel have the same slope. Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved. Problem 1 Checking for Parallel Lines Got It? Line <3 contains A(213, 6) and B(21, 2). Line <4 contains C(3, 6) and D(6, 7). Are <3 and <4 parallel? Explain. 12. To determine whether lines /3 and /4 are are parallel slope check whether the lines have the same 9. 13. Find the slope of each line. slope of <3 226 5 21 2 (213) slope of <4 24 12 5 726 623 2 13 14. Are the slopes equal? 5 13 Yes / No 15. Are lines /3 and /4 parallel? Explain. No. Explanations may vary. Sample: Since the slopes are _______________________________________________________________________ not equal, the lines are not parallel. _______________________________________________________________________ 87 Lesson 3-8 Writing Equations of Parallel Lines Problem 2 Got It? What is an equation of the line parallel to y 5 2x 2 7 that contains (25, 3)? 16. The slope of the line y 5 2x 2 7 is 21 . 17. The equation of the line parallel to y 5 2x 2 7 will have slope m 5 21 . 18. Find the equation of the line using point-slope form. Complete the steps below. m(x 2 x1) y 2 y1 5 y 2 3 5 21fx 2 (25)g Write in point-slope form. Substitute point and slope into equation. y235 2x 2 5 Simplify. y5 2x 2 2 Add 3 to both sides. Key Concept Slopes of Perpendicular Lines • If two nonvertical lines are perpendicular, then the product of their slopes is 21. • If the slopes of two lines have a product of 21, then the lines are perpendicular. • Any horizontal line and vertical line are perpendicular. F 19. The second bullet in the Take Note is the contrapositive of the first bullet. T 20. The product of the slopes of any horizontal line and any vertical line is 21. Problem 3 Checking for Perpendicular Lines Got It? Line <3 contains A(2, 7) and B(3,21). Line <4 contains C(22, 6) and D(8, 7). Are <3 and <4 perpendicular? Explain. 21. Find the slopes and multiply them. 28 27 m3 5 21 32 2 5 1 5 28 m3 3 m4 5 6 1 5 10 m4 5 8 722(22) 28 1 (28) Q 10 R 5 10 22. Underline the correct words to complete the sentence. Lines /3 and /4 are / are not perpendicular because the product of their slopes does / does not equal 21. Chapter 3 88 Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved. Write T for true or F for false. Problem 4 Writing Equations of Perpendicular Lines Got It? What is an equation of the line perpendicular to y 5 23x 2 5 that contains (23, 7)? 23. Complete the reasoning model below. Write Think I can identify the slope, m1, of y 5 23x 2 5 is in point-slope form, so m1 5 the given line. I know that the slope, m2, of m2 is the perpendicular line is 1 because 3 3 I can use m2 and (23, 7) 5 21. 3 3 the negative reciprocal of m1. 1 3 . y 2 y1 5 m(x 2 x1) to write the equation of 1 [x 2 (23)] 3 1 y 2 7 5 (x 1 3) 3 y275 the perpendicular line in point-slope form. Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved. Lesson Check • Do you UNDERSTAND? Error Analysis Your classmate tries to find an equation for a line parallel to y 5 3x 2 5 that contains (24, 2). What is your classmate’s error? 24. Parallel lines have the same / different slopes. 25. Show a correct solution in the box below. slope of given line 3 1 slope of parallel line 3 y y1 m(x x1) 1 y 2 (x 4) 3 slope of given line = 3; slope of parallel line = 3 y 2 2 5 3(x 1 4) Math Success Check off the vocabulary words that you understand. slope reciprocal parallel perpendicular Rate how well you understand perpendicular lines. Need to review 0 2 4 6 8 Now I get it! 10 89 Lesson 3-8