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Name Class 3-2 Date Angles Formed by Parallel Lines and Transversals Going Deeper Essential question: How can you prove and use theorems about angles formed by transversals that intersect parallel lines? 1 G-CO.3.9 Engage Introducing Transversals Recall that 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. t 1 2 p 4 3 5 6 8 q 7 Angle Pair Example 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 are angles that lie on opposite sides of the transversal outside the intersected lines. ∠2 and ∠8 © Houghton Mifflin Harcourt Publishing Company The following postulate is the starting point for proving theorems about parallel lines that are intersected by a transversal. Same-Side Interior Angles Postulate t If two parallel lines are cut by a transversal, then the pairs of same-side interior angles are supplementary. 1 3 4 Given p ǁ q, ∠4 and ∠5 are supplementary. Given p ǁ q, ∠3 and ∠6 are supplementary. 5 8 2 6 7 p q REFLECT 1a. Explain how you can find m∠3 in the postulate diagram if p ǁ q and m∠6 = 61°. 1b. In the postulate diagram, suppose p ǁ q and line t is perpendicular to line p. Can you conclude that line t is perpendicular to line q ? Explain. Chapter 3 95 Lesson 2 2 G-CO.3.9 proof Alternate Interior Angles Theorem t If two parallel lines are cut by a transversal, then the pairs of alternate interior angles have the same measure. 1 4 5 Given: p ǁ q Prove: m∠3 = m∠5 8 2 p 3 6 q 7 Complete the proof by writing the missing reasons. Choose from the following reasons. You may use a reason more than once. Same-Side Interior Angles Postulate Subtraction Property of Equality Given Definition of supplementary angles Substitution Property of Equality Statements Linear Pair Theorem Reasons 1. p ǁ q 1. 2. ∠3 and ∠6 are supplementary. 2. 3. m∠3 + m∠6 = 180° 3. 4. ∠5 and ∠6 are a linear pair. 4. 5. ∠5 and ∠6 are supplementary. 5. 6. m∠5 + m∠6 = 180° 6. 7. m∠3 + m∠6 = m∠5 + m∠6 7. 8. m∠3 = m∠5 8. © Houghton Mifflin Harcourt Publishing Company REFLECT 2a. Suppose m∠4 = 57° in the above figure. Describe two different ways to determine m∠6. 2b. In the above figure, explain why ∠1, ∠3, ∠5, and ∠7 all have the same measure. 2c. In the above figure, is it possible for all eight angles to have the same measure? If so, what is that measure? Chapter 3 96 Lesson 2 3 G-CO.3.9 proof Corresponding Angles Theorem t If two parallel lines are cut by a transversal, then the pairs of corresponding angles have the same measure. 1 4 Given: p ǁ q Prove: m∠1 = m∠5 2 p 3 5 6 8 7 q Complete the proof by writing the missing reasons. Statements Reasons 1. p ǁ q 1. 2. m∠3 = m∠5 2. 3. m∠1 = m∠3 3. 4. m∠1 = m∠5 4. REFLECT © Houghton Mifflin Harcourt Publishing Company 3a. Explain how you can you prove the Corresponding Angles Theorem using the Same-Side Interior Angles Postulate and a linear pair of angles. Many postulates and theorems are written in the form “If p, then q.” The converse of such a statement has the form “If q, then p.” The converse of a postulate or theorem may or may not be true. The converse of the Same-Side Interior Angles Postulate is accepted as true, and this makes it possible to prove that the converses of the previous theorems are true. Converse of the Same-Side Interior Angles Postulate If two lines are cut by a transversal so that a pair of same-side interior angles are supplementary, then the lines are parallel. Converse of the Alternate Interior Angles Theorem If two lines are cut by a transversal so that a pair of alternate interior angles have the same measure, then the lines are parallel. Converse of the Corresponding Angles Theorem If two lines are cut by a transversal so that a pair of corresponding angles have the same measure, then the lines are parallel. Chapter 3 97 Lesson 2 A paragraph proof is another way of presenting a mathematical argument. As in a two-column proof, the argument must flow logically and every statement should have a reason. 4 G-CO.3.9 proof Equal-Measure Linear Pair Theorem If two intersecting lines form a linear pair of angles with equal measures, then the lines are perpendicular. 1 2 Given: m∠1 = m∠2 Prove: ℓ ⊥ m m Complete the following paragraph proof. It is given that ∠1 and ∠2 form a linear pair. Therefore, ∠1 and ∠2 are supplementary by the . By the definition of supplementary angles, m∠1 + m∠2 = 180°. It is also given that m∠1 = m∠2. So, m∠1 + m∠1 = 180° by the . Simplifying gives 2m∠1 = 180° and m∠1 = 90° by the Division Property of Equality. Therefore, ∠1 is a right angle and ℓ ⊥ m by the . REFLECT 4a. State the converse of the Equal-Measure Linear Pair Theorem shown above. Is the converse true? In Exercises 1−2, complete each proof by writing the missing statements or reasons. 1. I f two parallel lines are cut by a transversal, then the pairs of alternate exterior angles have the same measure. t 1 4 iven: p ǁ q G Prove: m∠1 = m∠7 5 8 Statements q Reasons 1. p ǁ q 1. 2. m∠1 = m∠5 2. 3. m∠5 = m∠7 3. 4. m∠1 = m∠7 4. Chapter 3 p 3 6 7 2 98 Lesson 2 © Houghton Mifflin Harcourt Publishing Company practice 2. P rove the Converse of the Alternate Interior Angles Theorem. t 1 4 iven: m∠3 = m∠5 G Prove: p ǁ q 5 8 Statements p 6 7 q Reasons 1. m∠3 = m∠5 1. 2. ∠5 and ∠6 are a linear pair. 2. Definition of linear pair 3. 3. Linear Pair Theorem 4. m∠5 + m∠6 = 180° 4. 5. m∠3 + m∠6 = 180° 5. 6. ∠3 and ∠6 are supplementary. 6. 7. p ǁ q 7. m 3. Complete the paragraph proof. p 1 iven: ℓ ǁ m and p ǁ q G Prove: m∠1 = m∠2 © Houghton Mifflin Harcourt Publishing Company 2 3 2 3 q It is given that p ǁ q, so m∠1 = m∠3 by the . It is also given that ℓ ǁ m, so m∠3 = m∠2 by the . Therefore, m∠1 = m∠2 by the . 4. T he figure shows a given line m, a given point P, and the construction of a line ℓ that is parallel to line m. Explain why line ℓ is parallel to line m. P m Chapter 3 99 Lesson 2 5. Can you use the information in the figure to conclude that p ǁ q? Why or why not? 71˚ 1 p 2 3 4 109˚ 5 6 q © Houghton Mifflin Harcourt Publishing Company Chapter 3 100 Lesson 2 3-2 Name ________________________________________ Date __________________ Class__________________ Name LESSON Practice Class Date Angles Formed by Parallel Lines and Transversals Additional Practice 3-2 Find each angle measure. 1. m∠1 _______________________ 2. m∠2 _______________________ 3. m∠ABC _______________________ 4. m∠DEF _______________________ Complete the two-column proof to show that same-side exterior angles are supplementary. 5. Given: p || q Prove: m∠1 + m∠3 = 180° Proof: © Houghton Mifflin Harcourt Publishing Company Statements Reasons 1. p || q 1. Given 2. a. _______________________ 2. Lin. Pair Thm. 3. ∠1 ≅ ∠2 3. b. _______________________ 4. c. _______________________ 4. Def. of ≅ ∠s 5. d. _______________________ 5. e. _______________________ 6. Ocean waves move in parallel lines toward the shore. The figure shows Sandy Beaches windsurfing across several waves. For this exercise, think of Sandy’s wake as a line. m∠1 = (2x + 2y)° and m∠2 = (2x + y)°. Find x and y. x = _________ y = _________ Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor. Chapter 3 101 16 Lesson 2 Holt McDougal Geometry Name ________________________________________ Date __________________ Class__________________ Problem Solving Problem Solving LESSON 3-2 Angles Formed by Parallel Lines and Transversals Find each value. Name the postulate or theorem that you used to find the values. 2. In the diagram, roads a and b are parallel. 1. In the diagram of movie theater seats, the incline of the floor, f, is parallel to the seats, s. If m∠1 = 68°, what is x? What is the measure of ∠PQR? _________________________________________ ________________________________________ 3. In the diagram of the gate, the horizontal bars are parallel and the vertical bars are parallel. Find x and y. _________________________________________ _________________________________________ _________________________________________ Use the diagram of a staircase railing for Exercises 4 and 5. AG || CJ and AD || FJ . Choose the best answer. © Houghton Mifflin Harcourt Publishing Company 4. Which is a true statement about the measure of ∠DCJ? A It equals 30°, by the Alternate Interior Angles Theorem. B It equals 30°, by the Corresponding Angles Postulate. C It equals 50°, by the Alternate Interior Angles Theorem. D It equals 50°, by the Corresponding Angles Postulate. 5. Which is a true statement about the value of n? F It equals 25°, by the Alternate Interior Angles Theorem. G It equals 25°, by the Same-Side Interior Angles Theorem. H It equals 35°, by the Alternate Interior Angles Theorem. J It equals 35°, by the Same-Side Interior Angles Theorem. Chapter 3 Copyright © by Holt McDougal. Additions and changes102 Original content to the original content are the responsibility of the instructor. 100 Lesson 2 Holt McDougal Geometry