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LESSON 10.3 Parallel Lines and Perpendicular Lines LESSON AT A GLANCE F C R Focus: Common Core State Standards 4.G.A.1 Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in two dimensional figures. Mathematical Practices MP4 Model with mathematics. MP5 Use appropriate tools strategically. MP6 Attend to precision. F C R Coherence: Standards Across the Grades Before Grade 4 After 3.G.A.1 4.G.A.1 5.G.B.3 5.G.B.4 F C R Rigor: Level 1: Understand Concepts....................Share and Show ( Checked Items) Level 2: Procedural Skills and Fluency.......On Your Own Level 3: Applications..................................Think Smarter and Go Deeper Learning Objective Identify and draw parallel lines and perpendicular lines. Language Objective Students discuss with a partner which tools they need to identify and draw parallel lines and perpendicular lines. Materials MathBoard, straightedge F C R For more about how GO Math! fosters Coherence within the Content Standards and Mathematical Progressions for this chapter, see page 547J. About the Math Professional Development Why Teach This? Interactive Student Edition In this lesson, students learn to distinguish between intersecting lines and parallel lines in a plane: intersecting lines cross at exactly one point, while parallel lines never share a commonProfessional point. Parallel lines are always the same Development distance apart. Students also learn that perpendicular lines are intersecting lines that form four right angles. About the Math Personal Math Trainer Math on the Spot As students work through the lesson, be sure to relate the new terms to real-world examples. The language of geometry is key to describing things around us. Encourage students to draw and label figures described. Creating models of parallel, perpendicular, and intersecting lines will help students later when distinguishing between parallelograms and rectangles. Professional Development Videos 561A Chapter 10 Animated Math Models iTools: Geometry HMH Mega Math Daily Daily Routines Routines Common Core Common Core Problem of the Day 10.3 In baseball, home plate is a five-sided figure with five angles. What type of angle is angle A? right A E B D 1 ENGAGE with the Interactive Student Edition Essential Question How can you identify and draw parallel lines and perpendicular lines? Making Connections Invite students to tell you what they know about parking lots. What is a parking lot? A parking lot is usually a rectangular shaped area with lines for people to park their cars in. Learning Activity What is the problem the students are trying to solve? Connect the story to the problem. C Vocabulary intersecting lines, parallel lines, perpendicular lines • What problem are you trying to solve? Find out what kind of lines have been formed in the parking lot. • If the two lines that are side by side were made longer, would they ever cross? No, the lines always stay the same distance apart • Interactive Student Edition • Multimedia eGlossary • What kind of angles are formed where the line that runs along the row of cars meets with the lines that the cars are parked between? right angles Literacy and Mathematics Vocabulary Builder Materials Word Description (see eTeacher Resources) Word Description Have students use the Word Description graphic organizer to define the new vocabulary words in this lesson and any previous terms in this chapter with which they have difficulty. Drawing examples and nonexamples is important to distinguish between the different types of lines or angles. Write the definition. • Have students construct a miniature parking lot in the classroom. Then have students summarize their observations. • Have students discuss how a parking lot reflects the properties they know about pairs of parallel sides in rectangles. Describe using facts and characteristics. parallel lines Write or draw an example. Choose one or more of the following activities. How can you identi fy and draw parallel lines and perpendicular lines? Write or draw a nonexample. Lesson 10.3 561B 10.3 Unlock the Problem Parallel Lines and Perpendicular Lines To introduce the lesson, have students watch the Real World Video, Maglev Trains. Maglev is short for “magnetic levitation”. Levitation means to rise in the air. Unlock Unlock the the Problem Problem You can find models of lines in the world around you. For example, two streets that cross each other model intersecting lines. Metal rails on a train track that never cross model parallel lines. MP6 Attend to precision. Review the information in the chart. • How are intersecting lines and perpendicular lines the same? How are they different? Possible answer: both types of lines cross Term and Definition Try This! Explain that the phrase “appear to be” is used because it is not possible to tell whether the streets on the map are really parallel; if they were extended, they may intersect. • Can more than two lines be parallel? Explain. Yes; possible explanation: if all the lines are ‹__› F G ‹__› ‹__› Line DE is parallel to line FG. DE i FG The symbol i means “is parallel to.” ‹___› ‹___› Line LM is perpendicular LM ' NO to line NO. The symbol ' means “is perpendicular to.” N M O 37 th W Write perpendicular, parallel, or intersecting. 36 th W • W 36th St and Broadway intersecting _____ • W 35th St and 7th Ave intersecting, perpendicular _____ • W 37th St and W 36th St parallel _____ Math th Talk Yes; possible explanation: a ray is a part of a line. If the lines the rays are part of are parallel, then the rays are parallel. 35 St St St ay W Try This! Tell how the streets appear to be related. Broadw Draw on the board two parallel lines. •Use hand gestures as you tell students that parallel lines never intersect and never cross each other. I E L © Houghton Mifflin Harcourt Publishing Company • Image Credits: (t) ©Bernd Mellmann/Alamy Images Restate Write It ‹__› Line HI intersects line JK HI and JK intersect at at point X. point X J D Perpendicular lines are lines in a plane that intersect to form four right angles. intersect and parallel lines never intersect. Read It K X Parallel lines are lines in a plane that are always the same distance apart. Parallel lines never intersect. • Can parallel lines ever be perpendicular? Explain. No; possible explanation: perpendicular lines ▲ Maglev trains use magnets to lift them above the tracks while moving. Draw It H Intersecting lines are lines in a plane that cross at exactly one point. Intersecting lines form four angles. at one point, forming four angles. Only perpendicular lines form four angles that are right angles. Geometry— 4.G.A.1 MATHEMATICAL PRACTICES MP4, MP6, MP7 Essential Question How can you identify and draw parallel lines and perpendicular lines? MATHEMATICAL PRACTICES ELL Strategy: Lesson 10.3 Name Av e 2 EXPLORE 4.G.A.1 Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify DO NOTinEDIT--Changes must be made through “File info” these two-dimensional figures. CorrectionKey=B 7th LESSON MATHEMATICAL PRACTICES 6 Use Math Vocabulary Can two rays be parallel? Explain. Chapter 10 561 extended, it is possible that more than two of the lines will never touch, so those lines are all parallel. • Are all intersecting lines perpendicular? Explain. No; possible explanation: perpendicular lines are special types of intersecting lines. Some intersecting lines that form angles are not right angles. MP4 Model with mathematics. Have students make a diagram to help explain. • Two lines are perpendicular to a third line. How are the first two lines related to each other? Explain. Possible answer: the lines 4_MNLESE342286_C10L03.indd 561 Reteach 10.3 parallel. Use Math Talk to help students recognize that two rays can be • Suppose two lines are parallel. Can a part of each of these lines be a ray? Explain. 2 DO NOT EDIT--Changes must be made through "File info" CorrectionKey=A 1 Lesson 10.3 Reteach Name Alphabet Soup Use all 26 capital letters of the alphabet. Place them into as many “soups” as possible. Parallel lines are lines in a plane that are always the same distance apart. Parallel lines or line segments never meet. Possible answers are given. In the figure, lines AB and CD, even if extended, will never meet. The lines are parallel. Write ABiCD. Lines AD and BC are also parallel. So, ADiBC. B D C Lesson 10.3 Enrich Name Parallel Lines and Perpendicular Lines A Differentiated Instruction Enrich 10.3 3 DO NOT EDIT--Changes must be made through "File info" CorrectionKey=A are parallel. My drawing shows that the two lines are always the same distance apart. Math Talk 10/13/14 8:07 PM 1. Letters Intersecting lines cross at exactly one point. Intersecting lines that form right angles are perpendicular. In the figure, lines AD and AB are perpendicular because they form right angles at vertex A. Write AD'AB. with parallel line segments E, F, H, M, N, Z 2. Letters with perpendicular line segments E, F, H, L, T Lines BC and CD are also perpendicular. So, BC'CD. Use the figure for 1–3. 1. Name RV and ST 2. Name 3. Letters V two sides that appear to be parallel. two sides that appear to be perpendicular. RV and RS or RS and ST T with intersecting, but not perpendicular, line segments R two sides that appear to be intersecting, but not perpendicular. with no parallel, perpendicular, or intersecting line segments S 3. Name 4. Letters RV and VT or TV and ST A, K, M, N, V, W, X, Y, Z B, C, D, G, I, J, O, P, Q, R, S, U Possible answer: yes; the part of each line from a point on the line extending in one direction is a ray. 561 Chapter 10 Chapter Resources © Houghton Mifflin Harcourt Publishing Company 4_MNLEAN343153_C10R03.indd 9 10-9 Reteach Chapter Resources © Houghton Mifflin Harcourt Publishing Company 24/02/14 5:15 PM 4_MNLEAN343153_C10E03.indd 10 10-10 Enrich 24/02/14 5:02 PM DO NOT EDIT--Changes must be made through “File info” CorrectionKey=B Activity Draw and label YX ' YZ ___› ___› DO NOT EDIT--Changes must be made through “File info” CorrectionKey=B Activity Students draw two perpendicular rays to form right angle ZYX. • How are rays YX and YZ related? Possible intersecting at point Y. Materials ■ straightedge answer: they both have the same endpoint Y. ___› STEP 1: Draw and label YX . • How can you check if two rays are ___› STEP 2: Then draw and label YZ . I can use a square corner to check if Z Y • What does the upside-down T symbol mean? that the rays are perpendicular • What are all the ways to name the figure you drew? angle ZYX, angle XYZ, angle Y perpendicular? the rays are perpendicular. ___› X ___› STEP 3: Make sure YX and YZ intersect at point Y. 3 EXPLAIN STEP 4: Make sure the rays are perpendicular. 1. Name the figure you drew. ∠ZYX 2. Can you classify the figure? Explain. __› Share and Show __› ∠ZYX is a right angle, since YX and YZ are perpendicular, MATH BOARD ___ ___ 1. Draw and label QR i ST . Think: Parallel lines never intersect. Parallel line segments are parts of parallel lines. Q R S T Use Math Talk to focus on students’ understanding of the symbols for perpendicular and parallel lines. ___ C D B F CD and BF 3. Name two line segments that appear to be perpendicular. ___ ___ CD and BC Possible answer: the symbol i looks like parallel line segments, and the symbol ' looks like perpendicular line segments. Math Talk MATHEMATICAL PRACTICES 4 Use Symbols How could the symbols ' and i help you remember which relationships they describe? 562 © Houghton Mifflin Harcourt Publishing Company Possible answers are given. 2. Name two line segments that appear to be parallel. ___ Math Talk © Houghton Mifflin Harcourt Publishing Company Use the figure for 2 and 3. • What do the symbols for parallel and perpendicular look like? two parallel line segments; two perpendicular line segments If If 4_MNLESE342286_C10L03.indd 562 Quick Check Quick Check 3 2 1 3 2 1 Rt I Rt I a student misses the checked exercises 10/13/14 9:19 PM Then Advanced AdvancedLearners Learners BOARD The first problem connects to the learning model. Have students use the MathBoard to explain their thinking. Use the checked exercises for Quick Check. Students should show their answers for the Quick Check on the MathBoard. and perpendicular rays form right angles. Share Share and and Show Show MATH Visual / Kinesthetic Partners Materials ruler or straightedge •Have students draw street maps based on a description given by their partner. •One student starts by drawing a horizontal line across the middle of a blank sheet of paper and labeling it Main Street. •The partner then describes another street and its geometric relationship to Main Street, while the other student draws it. For example, “Oak Street is above and parallel to Main Street.” •Have the partner describe five streets that are either perpendicular, parallel, or intersecting to another street on the map. •Have partners switch roles and repeat. Differentiate Instruction with • Reteach 10.3 • Personal Math Trainer 4.G.A.1 • RtI Tier 1 Activity (online) COMMON ERRORS COMMON ERRORS Error Students may confuse perpendicular lines and parallel lines. Example For Exercise 2, students name sides CB and BF as parallel. Springboard to Learning Help students remember the meaning of parallel by underlining the pair of l’s in the word, which look like parallel lines. Lesson 10.3 562 Name On Your Own On On Your Your Own Own If students complete the checked exercises correctly, they may continue with the remaining exercises. In Exercises 4 and 5, students identify perpendicular lines and parallel lines in a geometric ﬁgure. Have students note that lines or sides of ﬁgures can appear to be parallel or perpendicular. Use the figure for 4–5. 4. Name a pair of lines that are perpendicular. ‹__› ‹___› 6. RS i TU C S T M U ‹___› ‹___› K L ‹___› 9. JK ' LM D E ‹___› ‹___› ‹___› 10. ST intersecting UV at 11. AB i FG point X J L S M SMARTER V X K Exercise 12 requires students to recognize that if the lines were extended, they would eventually intersect. • Are lines HL and IM parallel? Explain. No; 563 Chapter 10 H J Possible drawings are shown. ___› ____› ___ ___ 7. KL and KM 8. CD ' DE ___ ___ MATHEMATICAL PRACTICES the perpendicular line will intersect both parallel lines at right angles since the parallel lines are the same distance apart. B C Draw and label the figure described. Problem Solving • Applications U T A B F G OqnakdlRnkuhmf¤@ookhb`shnmr OqnakdlRnkuhmf¤@ookhb`shnmr Use the figure for 12–13. 12. © Houghton Mifflin Harcourt Publishing Company MP7 Look for and make use of structure. • If a line is perpendicular to one of two parallel lines, is it perpendicular to the other line as well? Explain. Yes; possible explanation: ‹__› DC and FJ 4 ELABORATE DEEPER G F A 5. Name a pair of lines that appear to be parallel. R possible explanation: the lines appear not to be the same distance apart. D ‹___› FJ and HG ‹___› SMARTER ‹___› Dan says that HL is parallel to IM. Is Dan correct? Explain. No; possible explanation: if you extend the lines, they G J K H L I M will intersect. 13. DEEPER Name two intersecting line segments that are not perpendicular. ___ ___ Possible answer: HL and LM Chapter 10 • Lesson 3 563 DO NOT EDIT--Changes must be made through “File info” CorrectionKey=B MATHEMATICAL PRACTICES ANALYZE • LOOK FOR STRUCTURE • PRECISION Use the house plan at the right for 14–16. SMARTER 14. What geometric term describes a corner Kitchen of the living room? Master Bedroom Living Room Possible answers: right angle or perpendicular line segments Family Room Dining Room 15. Name three parts of the plan that show line segments. Bedroom Bathroom Possible answers: a wall in the master bedroom, the For Exercise 16, students must find a pair of line segments that are always the same distance apart. In the case of these house plans, opposite walls of rooms and outer walls of the house appear to be parallel. bathroom door, a wall in the dining room 16. SMARTER Math on the Spot Video Tutor Name a pair of line segments that appear to be parallel. Use this video to help students model and solve this type of Think Smarter problem. Possible answer: two walls in the living room that do not intersect and are opposite each other Use the map at the right for 17–19. Math on the Spot videos are in the Interactive Student Edition and at www.thinkcentral.com. 17. Name a street that appears to be parallel to S 17th Street. t St stnu Che t tin S 0th St 9th MP4 Model with mathematics. In Exercises 17–19, students identify and name parallel and perpendicular streets on a map. • Why do you think city streets often are designed to be parallel or perpendicular? St St St Vernon St Use Diagrams Name a street that appears to be parallel to Vernon Street. 4 7th MATHEMATICAL PRACTICE 8th S1 18. S1 Street S1 Aus S2 Possible answers: S 18th Street, S 19th Street, S 20th Perry St Perry Street Possible answer: it is easier to navigate and to get from one place to another. 19. Name a street that appears to be perpendicular to S 19th Street. Possible answers: Austin Street, Chestnut Street 20. SMARTER Choose the labels to make a true statement. ‹___› IJ I G B ‹___› GH ‹__› A E EF is perpendicular to ‹___› AB ___ AE ‹___› C GH F H J D © Houghton Mifflin Harcourt Publishing Company SMARTER 564 4_MNLESE342286_C10L03.indd 564 3/9/15 3:27 PM DIFFERENTIATED INSTRUCTION INDEPENDENT ACTIVITIES Activities Picture Perfect! 5 EVALUATE Formative Assessment Essential Question Differentiated Centers Kit Activities Concentrate! Exercise 20 assesses students’ application of the concept of perpendicular lines. Students who choose lines AB and CD have chosen parallel lines instead of perpendicular lines. These students may have reversed the meanings of the two terms. Students who choose line GH may not understand the concept of parallel or perpendicular lines since this line is neither parallel nor perpendicular to any lines shown in the figure. Using the Language Objective Literature Skateboarding Takes Shape Reflect Have students discuss with a partner which tools they need to answer the Essential Question. How can you identify and draw parallel lines and perpendicular lines? Possible answer: I can use the definitions. Perpendicular lines are lines in a plane that intersect to form four right angles. Parallel lines are lines in a plane that are always the same distance apart; they never intersect. Students complete orange Activity Card 13 by matching the names of lines, angles, and twodimensional shapes to pictures. Students complete orange Activity Card 13 by matching the names of lines, angles, and twodimensional shapes to pictures. Students read about how plane figures and angles are used in skateparks. Math Journal WRITE Math Draw and label an example of two parallel lines that are perpendicular to a third line. Lesson 10.3 564 DO NOT EDIT--Changes must be made through “File info” CorrectionKey=B Practice and Homework Name Lesson 10.3 Parallel Lines and Perpendicular Lines COMMON CORE STANDARD—4.G.A.1 Draw and identify lines and angles, and classify shapes by properties of their lines and angles. Use the figure for 1–2. 1. Name a pair of lines that appear to be perpendicular. Practice and Homework Think: Perpendicular lines form right angles. ‹___› ‹___› AB and EF appear to form right angles. ‹___› ‹__› AB and EF Use the Practice and Homework pages to provide students with more practice of the concepts and skills presented in this lesson. Students master their understanding as they complete practice items and then challenge their critical thinking skills with Problem Solving. Use the Write Math section to determine student’s understanding of content for this lesson. Encourage students to use their Math Journals to record their answers. A C E F B 2. Name a pair of lines that appear to be parallel. D ‹___› AB and CD ‹___› Possible drawings are shown. Draw and label the figure described. ‹___› ‹___› ‹___› ‹____› ‹__› ‹___› 3. MN and PQ intersecting at 4. WX || YZ 5. FH ⊥ JK point R M W X Q R P N F J Y Z K H Problem Problem Solving Solving Use the street map for 6–7. 6. Name two streets that intersect but do not appear Maple to be perpendicular. Birch k Oa 7. Name two streets that appear to be parallel to each other. Elm © Houghton Mifflin Harcourt Publishing Company Oak and Maple or Oak and Birch Maple and Birch 8. Math Draw and label an example of WRITE two parallel lines that are perpendicular to a third line. Check students’ drawings. Chapter 10 4_MNLESE342286_C10P03.indd 565 565 Chapter 10 565 10/13/14 8:16 PM TEST PREP Lesson Check (4.G.A.1) 1. Write a capital letter that appears to have perpendicular line segments? 2. In the figure, which pair of line segments appear to be parallel? F J Possible answer: T Continue concepts and skills practice with Lesson Check. Use Spiral Review to engage students in previously taught concepts and to promote content retention. Common Core standards are correlated to each section. G H FG and JH Spiral Review (4.NBT.B.5, 4.NBT.B.6 , 4.NF.A.2, 4.G.A.2) acute angles did he draw? 2 acute angles 5. A school principal ordered 1,000 pencils. He gave an equal number to each of 7 teachers until he had given out as many as possible. How many pencils were left? 6 pencils 4. Mike drank more than half the juice in his glass. What fraction of the juice could Mike have drunk? _ Possible answer: 5 8 6. A carton of juice contains 64 ounces. Ms. Wilson bought 6 cartons of juice. How many ounces of juice did she buy? 384 ounces © Houghton Mifflin Harcourt Publishing Company 3. Nolan drew a right triangle. How many FOR MORE PRACTICE GO TO THE 566 Personal Math Trainer Lesson 10.3 566