4.3 Proving Lines are Parallel

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1 Nae Cass Date 4.3 Proving Lines are Parae Essentia Question: How can you prove that two ines are parae? Expore Writing Converses of Parae Line Theores You for the converse of and if-then stateent "if p, then q" by swapping p and q. The converses of the postuate and theores you have earned about ines cut by a transversa are true stateents. In the Expore, you wi write specific cases of each of these converses. Resource Locker The diagra shows two ines cut by a transversa t. Use the diagra and the given stateents in Steps D. You wi copete the stateents based on your work in Steps D. Stateents ines and are parae 4 6 and 3 are suppeentary t Use two of the given stateents together to copete a stateent about the diagra using the Sae-Side Interior nges Postuate. y the postuate: If ines and are parae, then 6 and 3 are suppeentary. Houghton Miffin Harcourt Pubishing Copany C D Now write the converse of the Sae-Side Interior nges Postuate using the diagra and your stateent in Step. y its converse: If 6 and 3 are suppeentary, then ines and are parae. Repeat to iustrate the ternate Interior nges Theore and its converse using the diagra and the given stateents. y the theore: If ines and are parae, then 4 6. y its converse: If 4 6, then ines and are parae. Use the diagra and the given stateents to iustrate the Corresponding nges Theore and its converse. y the theore: If ines and are parae, then 3 7. y its converse: if 3 7, then ines and are parae. Modue 4 85 Lesson 3 DO NOT EDIT--Changes ust be ade through "Fie info" CorrectionKey=NL-;C- Date

2 Refect. How do you for the converse of a stateent? Possibe answer: Reverse the hypothesis and concusion; for a stateent if p, then q, the converse is if q, then p.. What kind of anges are 4 and 6 in Step C? What does the converse you wrote in Step C ean? Possibe answer: aternate interior anges; if the two aternate interior anges 4 and 6 are congruent, then the ines and are parae. Expain Proving that Two Lines are Parae The converses fro the Expore can be stated foray as a postuate and two theores. (You wi prove the converses of the theores in the exercises.) Converse of the Sae-Side Interior nges Postuate If two ines are cut by a transversa so that a pair of sae-side interior anges are suppeentary, then the ines are parae. Converse of the ternate Interior nges Theore If two ines are cut by a transversa so that any pair of aternate interior anges are congruent, then the ines are parae. Converse of the Corresponding nges Theore If two ines are cut by a transversa so that any pair of corresponding anges are congruent, then the ines are parae. You can use these converses to decide whether two ines are parae. Exape osaic designer is using quadriatera-shaped coored ties to ake an ornaenta design. Each tie is congruent to the one shown here. The designer uses the coored ties to create the pattern shown here. Use the vaues of the arked anges to show that the two ines and are parae. Measure of : 0 Measure of : 60 Reationship between the two anges: They are suppeentary Houghton Miffin Harcourt Pubishing Copany Concusion: ǁ by the Converse of the Sae-Side Interior nges Postuate. Modue 4 86 Lesson 3

3 Now ook at this situation. Use the vaues of the arked anges to show that the two ines are parae. Measure of : 0 Measure of : 0 Reationship between the two anges: They are congruent corresponding anges. Concusion: ǁ by the Converse of the Corresponding nges Theore. Refect 3. What If? Suppose the designer had been working with this basic shape instead. Do you think the concusions in Parts and woud have been different? Why or why not? No, because the tie pattern fored sti has congruent corresponding ange and suppeentary ange pairs that can be used to produce parae ines. Your Turn Expain why the ines are parae given the anges shown. ssue that a tie patterns use this basic shape Houghton Miffin Harcourt Pubishing Copany = 0 and = 0 They are congruent aternate interior anges. The ines are parae because of the Converse of the ternate Interior nges Theore. = 0 and = 60 The anges are suppeentary. The ines are parae because of the Converse of the Sae-Side Interior nges Postuate. Modue 4 87 Lesson 3

4 Expain 3 Using nge Pair Reationships to Verify Lines are Parae When two ines are cut by a transversa, you can use reationships of pairs of anges to decide if the ines are parae. Exape 3 Use the given ange reationships to decide whether the ines are parae. Expain your reasoning. t 3 5 Houghton Miffin Harcourt Pubishing Copany Step Identify the reationship between the two anges. 3 and 5 are congruent aternate interior anges. Step re the ines parae? Expain. Yes, the ines are parae by the Converse of the ternate Interior nges Theore. 4 = (x + 0), 8 = (x + 5), and x = 5. Step Identify the reationship between the two anges. 4 = (x + 0) 8 = (x + 5) = ( + 0) = = ( + 5) = So, 4 and 8 are congruent corresponding anges. Step re the ines parae? Expain. Yes, the ines are parae by the Converse of the Corresponding nges Theore. Modue 4 89 Lesson 3

5 Your Turn Identify the type of ange pair described in the given condition. How do you know that ines and are parae? = 80 sae side interior anges; by the Converse of the Sae Side Interior nges Postuate t 9. 6 corresponding anges; by the Converse of the Corresponding nges Theore Eaborate 0. How are the converses in this esson different fro the postuate/theores in the previous esson? In the previous esson, we knew ines were parae and things about anges; here, we know things about ange pairs, and ines are parae.. What If? Suppose two ines are cut by a transversa such that aternate interior anges are both congruent and suppeentary. Describe the ines. The ines are parae and a the anges are 90. The transversa is perpendicuar to the ines.. Essentia Question Check-In Nae two ways to test if a pair of ines is parae, using the interior anges fored by a transversa crossing the two ines. Possibe answer: Use given inforation or easure pairs of anges to decide if aternate interior anges are congruent or if sae-side interior anges are suppeentary. Evauate: Hoework and Practice The diagra shows two ines cut by a transversa t. Use the diagra and the given stateents in Exercises 3 on the facing page. Stateents ines and are parae = 80 5 t Onine Hoework Hints and Hep Extra Practice Houghton Miffin Harcourt Pubishing Copany 4 6 Modue 4 90 Lesson 3

6 Expain Proving the Perpendicuar isector Theore Using Refections You can use refections and their properties to prove a theore about perpendicuar bisectors. These theores wi be usefu in proofs ater on. Perpendicuar isector Theore If a point is on the perpendicuar bisector of a segent, then it is equidistant fro the endpoints of the segent. Exape Prove the Perpendicuar isector Theore. Given: P is on the perpendicuar bisector of _. Refect Prove: P = P P Consider the refection across ine. Then the refection of point P across ine is aso P because point P ies on ine, which is the ine of refection. so, the refection of point across ine is by the definition of refection. Therefore, P = P because refection preserves distance. 3. Discussion What concusion can you ake about KLJ in the diagra using the Perpendicuar isector Theore? K M L JK = JL because point J ies on the perpendicuar bisector of _ KL. Houghton Miffin Harcourt Pubishing Copany J Modue 4 96 Lesson 4

7 Your Turn Use the diagra shown. _ D is the perpendicuar bisector of C. D 4. Suppose ED = 6 c and D = 0 c. Find DC. E ecause D is the perpendicuar bisector of C, then D = DC and DC = 0 c. C 5. Suppose EC = 5 c and = 5 c. Find C. ecause D is the perpendicuar bisector of C, then = C and C = 5 c. Expain Proving the Converse of the Perpendicuar isector Theore The converse of the Perpendicuar isector Theore is aso true. In order to prove the converse, you wi use an indirect proof and the Pythagorean Theore. In an indirect proof, you assue that the stateent you are trying to prove is fase. Then you use ogic to ead to a contradiction of given inforation, a definition, a postuate, or a previousy proven theore. You can then concude that the assuption was fase and the origina stateent is true. a c a + b = c Reca that the Pythagorean Theore states that for a right triange with egs of ength a and b and a hypotenuse of ength c, a + b = c. b Converse of the Perpendicuar isector Theore If a point is equidistant fro the endpoints of a segent, then it ies on the perpendicuar bisector of the segent. Houghton Miffin Harcourt Pubishing Copany Exape Prove the Converse of the Perpendicuar isector Theore Given: P = P Prove: P is on the perpendicuar bisector of _. Step : ssue what you are trying to prove is fase. ssue that P is not on the perpendicuar bisector of. Q Then, when you draw a perpendicuar ine fro P to the ine containing and, it intersects _ at point Q, which is not the idpoint of _. Step : Copete the foowing to show that this assuption eads to a contradiction. _ PQ fors two right trianges, QP and QP. So, Q + Q P = P and Q + Q P = P by the Pythagorean Theore. Subtract these equations: Q + QP = P Q + QP = P Q - Q = P - P However, P - P = 0 because P = P. Therefore, Q - Q = 0. This eans that Q = Q and Q = Q. This contradicts the fact that Q is not the idpoint of _. Thus, the initia assuption ust be incorrect, and P ust ie on the perpendicuar bisector of _. P Modue 4 97 Lesson 4

8 Refect 6. In the proof, once you know Q = Q, why can you concude that Q = Q? Take the square root of both sides. Since distances are nonnegative, Q = Q. Your Turn 7. _ D is 0 inches ong. _ D is 6 inches ong. Find the ength of _ C. D Since D is equidistant fro and C and D is perpendicuar to C by the diagra, then D ust be the perpendicuar bisector of C and C = C. C + 6 = 0, so C = 8 in. C and C = 6 in. Expain 3 Proving Theores about Right nges The sybo eans that two figures are perpendicuar. For exape, or XY _. Exape 3 Prove each theore about right anges. If two ines intersect to for one right ange, then they are perpendicuar and they intersect to for four right anges. Given: = 90 Prove: = 90, 3 = 90, 4 = 90 Stateent. = 90. Given. and are a inear pair.. Given 3. and are suppeentary. 3. Linear Pair Theore Reason 4. + = Definition of suppeentary anges = Substitution Property of Equaity 6. = Subtraction Property of Equaity 7. = 4 7. Vertica nges Theore 8. 4 = Substitution Property of Equaity 9. = 3 9. Vertica nges Theore 0. 3 = Substitution Property of Equaity If two intersecting ines for a inear pair of anges with equa easures, then the ines are perpendicuar. Given: = Prove: y the diagra, and for a inear pair so and are suppeentary by the Linear Pair Theore. y the definition of suppeentary anges, + = 80. It is aso given that =, so + = 80 by the Substitution Property of Equaity. dding gives = 80, and = 90 by the Division Property of Equaity. Therefore, is a right ange and by the definition of perpendicuar ines. Modue 4 98 Lesson 4 Houghton Miffin Harcourt Pubishing Copany

9 Refect 8. State the converse of the theore in Part. Is the converse true? If two intersecting ines are perpendicuar, then they for a inear pair of anges with equa easures; yes. Your Turn 9. Given: b ǁ d, c ǁ e, = 50, and 5 = 90. Use the diagra to find 4. d b a c e 4 = 40 ; by corresponding anges because c ǁ e, =, and by vertica anges, = 3, so 3 = 50 ; because 5 = 90, then a d and = 90, so 4 = 40. Eaborate 0. Discussion Expain how the converse of the Perpendicuar isector Theore justifies the copass-and-straightedge construction of the perpendicuar bisector of a segent. C D Houghton Miffin Harcourt Pubishing Copany The construction invoves aking two arcs that intersect in two points. Each of these two intersection points is equidistant fro the endpoints of the segent, because the arcs are the sae radius. So, both of the intersection points are on the perpendicuar bisector of the segent.. Essentia Question Check-In How can you construct perpendicuar ines and prove theores about perpendicuar bisectors? Constructing a ine perpendicuar to a given ine invoves using a copass to ocate two points that are not on the given ine but are equidistant fro two points on the given ine. You can prove the Perpendicuar isector Theore using a refection and its properties, and you can prove the Converse of the Perpendicuar isector Theore using an indirect arguent invoving the Pythagorean Theore. Modue 4 99 Lesson 4