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LESSON 4.2 Name Transversals and Parallel Lines 4.2 Date Transversals and Parallel Lines Essential Question: How can you prove and use theorems about angles formed by transversals that intersect parallel lines? Resource Locker Common Core Math Standards Explore The student is expected to: COMMON CORE Class G-CO.C.9 Exploring Parallel Lines and Transversals A transversal is a line that intersects two coplanar lines at two different points. In the figure, line t is a transversal. The table summarizes the names of angle pairs formed by a transversal. Prove theorems about lines and angles. Mathematical Practices COMMON CORE MP.3 Logic t 1 2 4 3 5 6 8 7 Language Objective Explain to a partner how to identify the angles formed by two parallel lines cut by a transversal. ENGAGE View the Engage section online. Discuss the photo. Ask students to describe ways that a real-world example of parallel lines and a transversal like this example differ from parallel lines and a transversal that might appear in a geometry book. Then preview the Lesson Performance Task. Corresponding angles lie on the same side of the transversal and on the same sides of the intersected lines. ∠ 1 and ∠5 Same-side interior angles lie on the same side of the transversal and between the intersected lines. ∠3 and ∠6 Alternate interior angles are nonadjacent angles that lie on opposite sides of the transversal between the intersected lines. ∠3 and ∠ 5 Alternate exterior angles lie on opposite sides of the transversal and outside the intersected lines. ∠1 and ∠ 7 Recall that parallel lines lie in the same plane and never intersect. In the figure, line ℓ is parallel to line m, written ℓǁm. The arrows on the lines also indicate that they are parallel. ℓ m ℓ‖ m Module 4 be ges must EDIT--Chan DO NOT Key=NL-C;CA-C Correction Lesson 2 175 gh “File info” made throu Date Class lel Lines ls and Paral Name Transversa 4.2 s formed angle ms about use theore prove and el lines? can you ect parall ion: How that inters transversals angles. lines and ms about HARDCOVER PAGES 153160 by Resource Locker Quest Essential COMMON CORE Para Exploring In the figure, sals nt points. ersal. Transver two differe d by a transv ar lines at Explore GE_MNLESE385795_U2M04L2.indd 175 and llel Lines theore G-CO.C.9 Prove two coplan of angle pairs intersects the names is a line that summarizes A transversal ersal. The table transv line t is a forme t 2 Turn to these pages to find this lesson in the hardcover student edition. p 1 4 3 5 6 8 7 q Example ∠ 1 and ∠5 Pair Angle on the same ersal and of the transv same side s lie on the ing angle between Correspondintersected lines. ersal and of the transv sides of the same side s lie on the interior angle ite sides of Same-side lie on oppos cted lines. angles that the interse jacent s are nonad lines. interior anglethe intersected outside Alternate ersal and between transv ersal the the transv ite sides of on oppos angles lie exterior Alternate cted lines. the interse and never same plane written ℓǁm. lie in the l to line m, l. parallel lines line ℓ is parallethat they are paralle Recall that figure, te the In ℓ‖ m also indica intersect. on the lines The arrows Credits: PREVIEW: LESSON PERFORMANCE TASK ∠3 and ∠6 ∠3 and ∠ 5 ∠1 and ∠ 7 ℓ m Lesson 2 175 Module 4 Lesson 4.2 4L2.indd 95_U2M0 ESE3857 GE_MNL 175 q Example y • Image g Compan Publishin Harcour t utterstock n Mifflin © Houghto Photographer/Sh ©Ruud Morijn Possible answer: Start by establishing a postulate about certain pairs of angles, such as same-side interior angles. The postulate allows you to prove a theorem about other pairs of angles, such as alternate interior angles. You can then use the postulate and the theorem to prove other theorems about other pairs of angles, such as corresponding angles. © Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Ruud Morijn Photographer/Shutterstock Essential Question: How can you prove and use theorems about angles formed by transversals that intersect parallel lines? Angle Pair p 175 13/06/15 11:06 PM 6/9/15 1:04 AM When parallel lines are cut by a transversal, the angle pairs formed are either congruent or supplementary. The following postulate is the starting point for proving theorems about parallel lines that are intersected by a transversal. EXPLORE Same-Side Interior Angles Postulate Exploring Parallel Lines and Transversals If two parallel lines are cut by a transversal, then the pairs of same-side interior angles are supplementary. Follow the steps to illustrate the postulate and use it to find angle measures. A t 1 2 4 3 5 6 8 7 B INTEGRATE TECHNOLOGY Draw two parallel lines and a transversal, and number the angles formed from 1 to 8. The properties of parallel lines and transversals can be explored using geometry software. Students can display lines and angle measures on screen, rotate a line so it is parallel to another, and observe the relationships between angles. p q Identify the pairs of same-side interior angles. ∠4 and ∠5; ∠3 and ∠6 C QUESTIONING STRATEGIES What does the postulate tell you about these same-side interior angle pairs? When two lines are cut by a transversal, how many pairs of corresponding angles are formed? How many pairs of same-side angles? 4; 2 Given p ‖ q, then ∠4 and ∠5 are supplementary and ∠3 and ∠6 are supplementary. D If m∠4 = 70°, what is m∠5? Explain. m∠5 = 110°; ∠4 and ∠5 are supplementary, so m∠4 + m∠5 = 180°. Therefore What does the Same-Side Interior Angles Postulate tell you about the measure of a pair of same-side interior angles? The sum of their measures is 180˚̊. 70° + m∠5 = 180°, so m∠5 = 110°. Reflect Explain how you can find m∠3 in the diagram if p ‖ q and m∠6 = 61°. ∠3 and ∠6 are supplementary, so m∠3 + m∠6 = 180°. Therefore m∠3 + 61° = 180°, so m∠3 = 119°. 2. What If? If m ‖ n, how many pairs of same-side interior angles are shown in the figure? What are the pairs? Two pairs; ∠3 and ∠5, ∠4 and ∠6 Module 4 176 m n 1 3 5 7 t 2 4 6 8 © Houghton Mifflin Harcourt Publishing Company 1. Lesson 2 PROFESSIONAL DEVELOPMENT GE_MNLESE385795_U2M04L2.indd 176 Math Background 6/9/15 1:04 AM When two parallel lines are cut by a transversal, alternate interior angles have the same measure, corresponding angles have the same measure, and same-side interior angles are supplementary. One of these three facts must be taken as a postulate and then the other two may be proved. In the work text, the statement about same-side interior angles is taken as the postulate. This is closely related to one of the postulates stated in Euclid’s Elements: “If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.” Transversals and Parallel lines 176 Explain 1 EXPLAIN 1 Proving that Alternate Interior Angles are Congruent Other pairs of angles formed by parallel lines cut by a transversal are alternate interior angles. Proving that Alternate Interior Angles are Congruent Alternate Interior Angles Theorem If two parallel lines are cut by a transversal, then the pairs of alternate interior angles have the same measure. CONNECT VOCABULARY To prove something to be true, you use definitions, properties, postulates, and theorems that you already know. Have students experience the idea of what the term alternate means by shading in alternating figures in a series of figures. Example 1 Prove the Alternate Interior Angles Theorem. Given: p ‖ q Prove: m∠3 = m∠5 t 1 2 4 3 5 6 8 7 QUESTIONING STRATEGIES p q Complete the proof by writing the missing reasons. Choose from the following reasons. You may use a reason more than once. Do you have to write a separate proof for every pair of alternate interior angles in the figure? Why or why not? No, you can write the proof for any pair of alternate interior angles. The same reasoning applies to all the pairs. • Same-Side Interior Angles Postulate • Subtraction Property of Equality • Given • Substitution Property of Equality • Definition of supplementary angles • Linear Pair Theorem © Houghton Mifflin Harcourt Publishing Company Statements Reasons 1. p ‖ q 1. Given 2. ∠3 and ∠6 are supplementary. 2. Same-Side Interior Angles Postulate 3. m∠3 + m∠6 = 180° 3. Definition of supplementary angles 4. ∠5 and ∠6 are a linear pair. 4. Given 5. ∠5 and ∠6 are supplementary. 5. Linear Pair Theorem 6. m∠5 + m∠6 = 180° 6. Definition of supplementary angles 7. m∠3 + m∠6 = m∠5 + m∠6 7. Substitution Property of Equality 8. m∠3 = m∠5 8. Subtraction Property of Equality Reflect 3. In the figure, explain why ∠1, ∠3, ∠5, and ∠7 all have the same measure. m∠1 = m∠3 and m∠5 = m∠7 (Vertical Angles Theorem), m∠3 = m∠5 (Alternate Interior Angles Theorem), so m∠1 = m∠3 = m∠5 = m∠7 (Transitive Property of Equality). Module 4 177 Lesson 2 COLLABORATIVE LEARNING GE_MNLESE385795_U2M04L2 177 Peer-to-Peer Activity Have students use lined paper or geometry software to draw two parallel lines and a transversal that is not perpendicular to the lines. Instruct the student’s partner to shade or mark the acute angles with one color and the obtuse angles with another color. Let students use a protractor or geometry software to see that all the angles with the same color are congruent, and that pairs of angles with different colors are supplementary. 177 Lesson 4.2 10/14/14 7:01 PM 4. Suppose m∠4 = 57° in the figure shown. Describe two different ways to determine m∠6. By the Alternate Interior Angles Theorem, m∠6 = 57°. Also ∠4 and ∠5 are supplementary, EXPLAIN 2 so m∠5 = 123°. Since ∠5 and ∠6 are supplementary, m∠6 = 57°. Explain 2 Proving that Corresponding Angles are Congruent Proving that Corresponding Angles are Congruent INTEGRATE MATHEMATICAL PRACTICES Focus on Reasoning MP.2 Explaining and justifying arguments is at the Two parallel lines cut by a transversal also form angle pairs called corresponding angles. Corresponding Angles Theorem If two parallel lines are cut by a transversal, then the pairs of corresponding angles have the same measure. Example 2 heart of this proof-based lesson. You may wish to have students pair up with a “proof buddy.” Students can exchange their work with this partner and check that the partner’s logical arguments make sense. Complete a proof in paragraph form for the Corresponding Angles Theorem. Given: p‖q t 1 2 4 3 Prove: m∠4 = m∠8 5 6 8 7 p q QUESTIONING STRATEGIES By the given statement, p‖q. ∠4 and ∠6 form a pair of alternate interior angles . m∠4 = m∠6 So, using the Alternate Interior Angles Theorem, What can you use as reasons in a proof? given information, properties, postulates, and previously-proven theorems . ∠6 and ∠8 form a pair of vertical angles. So, using the Vertical Angles Theorem, m∠6 = m∠8 . Using the Substitution Property of Equality in m∠4 = m∠6, substitute m∠4 for m∠6. The result is m∠4 = m∠8 © Houghton Mifflin Harcourt Publishing Company Reflect 5. Use the diagram in Example 2 to explain how you can prove the Corresponding Angles Theorem using the Same-Side Interior Angles Postulate and a linear pair of angles. By the Same-Side Interior Angles Theorem, m∠4 + m∠5 = 180°. As a linear pair, m∠4 + m∠1 = 180°. Therefore m∠4 + m∠1 = m∠4 + m∠5 , so m∠1 = m∠5. 6. How can you check that the first and last statements in a two-column proof are correct? The first statement should match the “Given” and the last statement should match the “Prove.” . Suppose m∠4 = 36°. Find m∠5. Explain. m∠5 = 144°; Since ∠1 and ∠4 form a linear pair, they are supplementary. So, INTEGRATE MATHEMATICAL PRACTICES Focus on Modeling MP.4 Draw parallel lines and a transversal on a m∠1 = 144°. Using the Corresponding Angles Theorem, you know that m∠1 = m∠5. So, m∠5 = 144°. Module 4 178 transparency. Trace an acute and an obtuse angle formed by the lines onto another transparency, and use them to find congruent angles. Lesson 2 DIFFERENTIATE INSTRUCTION GE_MNLESE385795_U2M04L2 178 Kinesthetic Experience Have students draw a pair of parallel lines and a transversal on translucent paper squares. Tear the paper between the parallel lines, and overlay the two parts to show that the angles are congruent. AVOID COMMON ERRORS 22/05/14 2:20 PM Some students may have difficulty identifying the correct angles for two parallel lines cut by a transversal because they are unfamiliar with the angles. Have these students use highlighters to color-code the different angle pairs. For example, students can use a yellow highlighter to highlight the term corresponding angles and then use the same highlighter to mark the corresponding angles. Transversals and Parallel lines 178 Explain 3 EXPLAIN 3 You can apply the theorems and postulates about parallel lines cut by a transversal to solve problems. Using Parallel Lines to Find Angle Pair Relationships Example 3 INTEGRATE MATHEMATICAL PRACTICES Focus on Modeling MP.4 Have students look through magazines to find In the diagram, roads a and b are parallel. Explain how to find the measure of ∠VTU. In the diagram, roads a and b are parallel. Explain how to find the measure of m∠WUV. It is given that m∠PRS = (9x)° and m∠WUV = (22x + 25)°. m∠PRS = m∠RUW by the Students may incorrectly apply the postulates and theorems presented in this lesson when lines cut by a transversal are not parallel. Remind them that the postulates and theorems are only true for parallel lines. QUESTIONING STRATEGIES Postulates may be used to prove theorems; which postulate was used to prove the two theorems in this lesson? Same-Side Interior Angles Postulate So, m∠RUW + m∠WUV = and x= 5 7. P S R V (2x - 22)° a Q P (9x)° U R S U T b V (22x + 25)° W 180° . Solving for x, 31x + 25 = 180, = 135° . In the diagram of a gate, the horizontal bars are parallel and the vertical bars are parallel. Find x and y. Name the postulates and/or theorems that you used to find the values. 126° 36° (12x + 2y)° (3x + 2y)° x = 10, y = 3; (12x + 2y)° = 126° by the Corresponding Angles Theorem and (3x + 2y)° = 36° by the Alternate Interior Angles Theorem. Solving the equations simultaneously results in x = 10 and y = 3. Module 4 Lesson 2 179 LANGUAGE SUPPORT GE_MNLESE385795_U2M04L2 179 5/22/14 9:20 PM Visual Cues Have students work in pairs to add the following angle definitions and pictures to an organizer like the one below: Angle(s) Alternate interior angles Same-side interior angles Lesson 4.2 Q T (x + 40)° Your Turn Corresponding angles 179 b .Substitute the value of x to find m∠WUV ; m∠WUV = (22(5) + 25)° © Houghton Mifflin Harcourt Publishing Company AVOID COMMON ERRORS Corresponding Angles Theorem . a ∠RUW and ∠WUV are supplementary angles. QUESTIONING STRATEGIES ELABORATE Find each value. Explain how to find the values using postulates, theorems, and algebraic reasoning. It is given that m∠PRQ = (x + 40)° and m∠VTU = (2x - 22)°. m∠PRQ = m∠RTS by the Corresponding Angles Theorem and m∠RTS = m∠VTU by the Vertical Angles Theorem. So, m∠PRQ = m∠VTU, and x + 40 = 2x - 22. Solving for x, x + 62 = 2x, and x = 62. Substitute the value of x to find m∠VTU: m∠VTU = (2(62) - 22)° = 102°. pictures with parallel lines and transversals, such as bridges, fences, furniture, etc. Use markers or colored tape to mark the lines, and then identify angles that appear to be congruent and angles that appear to be supplementary. How can you check that the postulates and theorems about parallel lines apply to realworld situations? Sample answer: Measure some angle pair relationships with lines that appear parallel, and verify that the postulates and theorems apply. Using Parallel Lines to Find Angle Pair Relationships Definition Picture Elaborate 8. SUMMARIZE THE LESSON How is the Same-Side Interior Angles Postulate different from the two theorems in the lesson (Alternate Interior Angles Theorem and Corresponding Angles Theorem)? The postulate shows that pairs of angles are supplementary, while the theorems show that Have students use the diagram to identify the following: pairs of angles have the same measure. 9. Discussion Look at the figure below. If you know that p and q are parallel, and are given one angle measure, can you find all the other angle measures? Explain. t 1 2 4 3 p 5 6 8 7 1 2 5 6 q 3 4 7 8 t Yes; Possible explanation: Consider angles 1–4. If you knew one angle you can use the p fact that angles that are linear pairs are supplementary and the Vertical Angles Theorem. q You could use either the Alternate Interior Angles Theorem or the Corresponding Angles Theorem to find one of the angle measures for angles 5–8. Then you can use the Linear Parallel lines p and q Pair Theorem and the Vertical Angles Theorem to find all of those angle measures. Transversal t Congruent angles: Corresponding ∠1 ≅ ∠3; ∠2 ≅ ∠4; ∠5 ≅ ∠7; ∠6 ≅ ∠8 10. Essential Question Check-In Why is it important to establish the Same-Side Interior Angles Postulate before proving the other theorems? You need to use the Same-Side Interior Angles Postulate to prove the Alternate Interior Alternate interior ∠2 ≅ ∠7; ∠3 ≅ ∠6 Angles Theorem, and to prove the Corresponding Angles Theorem you need to use either Supplementary angles: Same-side interior ∠2 and ∠3; ∠6 and ∠7 the Same-SIde Interior Angles Postulate or the Alternate Interior Angles Theorem. Evaluate: Homework and Practice In the figure below, m‖n. Match the angle pairs with the correct label for the pairs. Indicate a match by writing the letter for the angle pairs on the line in front of the corresponding labels. A. ∠4 and ∠6 C Corresponding Angles B. ∠5 and ∠8 A Same-Side Interior Angles C. ∠2 and ∠6 D Alternate Interior Angles D. ∠4 and ∠5 B m 1 3 5 7 t 2 4 6 8 Depth of Knowledge (D.O.K.) COMMON CORE Mathematical Practices 1 1 Recall of Information MP.2 Reasoning 2 1 Recall of Information MP.3 Logic 3 1 Recall of Information MP.2 Reasoning 2 Skills/Concepts MP.2 Reasoning 15 3 Strategic Thinking MP.4 Modeling 16 2 Skills/Concepts MP.4 Modeling 17 2 Skills/Concepts MP.3 Logic 4–14 EVALUATE ASSIGNMENT GUIDE Lesson 2 180 Exercise • Online Homework • Hints and Help • Extra Practice Vertical Angles Module 4 GE_MNLESE385795_U2M04L2.indd 180 n © Houghton Mifflin Harcourt Publishing Company 1. 20/03/14 9:47 AM Concepts and Skills Practice Explore Exploring Parallel Lines and Transversals Exercises 1–4 Example 1 Proving that Alternate Interior Angles are Congruent Exercises 5–14 Example 2 Proving that Corresponding Angles are Congruent Exercises 5–14 Example 3 Using Parallel Lines to Find Angle Pair Relationships Exercises 15–18 Transversals and Parallel lines 180