Lesson 3-3 Proving Parallel Lines (page 83) Essential Question How can you apply parallel lines (planes) to make deductions?

Proving Parallel Lines Recall from lesson 3-2 how we developed conclusions about pairs of angles given parallel lines. In this lesson we reverse our conclusions. We will look at the converses of the postulate and two theorems in lesson 3-2.

Postulate 11 corr. ∠’s ≅ ⇒ || - lines Converse of Postulate 10 If two lines are cut by a transversal & corresponding angles are congruent, then the lines are parallel. corr. ∠’s ≅ ⇒ || - lines x 1 2 4 3 5 6 y 8 7

Theorem 3-5 AIA ≅ ⇒ || - lines Converse of Theorem 3-2 If two lines are cut by a transversal and alternate interior angles are congruent, then the lines are parallel. AIA ≅ ⇒ || - lines t Given: ∠1 ≅ ∠ 2 transversal t cuts k & n Prove: k || n k 3 2 n 1

k || n Corr. ∠’s ≅ ⇒ || - lines ∠1 ≅ ∠ 2 Given ∠2 ≅ ∠ 3 Vert. ∠’s R ≅ transversal t cuts k & n Prove: k || n Proof: Statements Reasons ____________________________________ _____________________________________________ k 3 2 n 1 See page 83! ∠1 ≅ ∠ 2 Given ∠2 ≅ ∠ 3 Vert. ∠’s R ≅ ∠1 ≅ ∠ 3 Transitive Property k || n Corr. ∠’s ≅ ⇒ || - lines

Theorem 3-6 SSIA Supp. ⇒ || - lines Converse of Theorem 3-3 If two lines are cut by a transversal & same side interior angles are supplementary, then the lines are parallel. SSIA Supp. ⇒ || - lines t Given: transversal t cuts k and n ∠1 is supplementary to ∠2 Prove: k || n k 1 n 2 3

k || n AIA ≅⇒ || - lines Given m∠2 + m ∠ 3 = 180º ∠- Addition Post. Given: transversal t cuts k and n ∠1 is supplementary to ∠2 Prove: k || n Proof: k 1 n 2 3 Statements Reasons ________________________________________ _____________________________________________ ∠1 is supplementary to ∠ 2 Given m∠2 + m ∠ 3 = 180º ∠- Addition Post. ∠3 is supplementary to ∠ 2 Def. of Supp. ∠’s ∠1 ≅ ∠ 3 Supp. of same ∠ R ≅ k || n AIA ≅⇒ || - lines

In a plane, two lines perpendicular to the same line are parallel. Theorem 3-7 In a plane, two lines perpendicular to the same line are parallel. t Given: k ⊥ t n ⊥ t Prove: k || n k 1 n 2

k || n n ⊥ t k ⊥ t ; n ⊥ t Given Def. of ⊥ lines Given: k ⊥ t n ⊥ t Prove: k || n Proof: k 1 n 2 Statements Reasons ________________________________________ _____________________________________________ ________________________________________ _____________________________________________ k ⊥ t ; n ⊥ t Given m∠1 = 90º; m∠2 = 90º Def. of ⊥ lines m∠1 = m∠2; ∠1 ≅ ∠2 Substitution Property k || n Corr. ∠’s ≅ ⇒ || - lines

Theorem 3-8 Through a point outside a line, there is exactly one line parallel to the given line. Exactly one line means that the line exists and is unique. P. n k

Theorem 3-9 Through a point outside a line, there is exactly one line perpendicular to the given line. h P . Exactly one line means that the line exists and is unique. k

Two lines parallel to a third line are parallel to each other. Theorem 3-10 Two lines parallel to a third line are parallel to each other. k l n Given: k || l k || n Prove: l || n

Please take note: Theorem 3-10 is true for lines in space , however, Theorems 3-8 and 3-9 are true only for lines in the same plane .

Ways to Prove Two Lines Parallel: Show that a pair of corresponding angles are congruent. Show that a pair of alternate interior angles are congruent. Show that a pair of same-side interior angles are supplementary. In a plane show that both lines are perpendicular to a third line. Show that both lines are parallel to a third line.

Ways to Prove Two Lines Parallel: YOU MUST REMEMBER THIS, SO PUT IT ON YOUR CARD! Show that a pair of corresponding angles are congruent. Show that a pair of alternate interior angles are congruent. Show that a pair of same-side interior angles are supplementary. In a plane show that both lines are perpendicular to a third line. Show that both lines are parallel to a third line.

Example #1. Find the value of “x” that makes a || b, Example #1. Find the value of “x” that makes a || b, then find the value of “y” that makes m || n. m 12 (5x + 13)º 73º 5(12) + 13 (6x + 1)º 73º 6(12) + 1 (60 + 13)º (72 + 1)º x = ______ y = ______ 13 8(13) + 3 (104 + 3)º 107º (8y + 3)º n WHY? WHY? a b If 5 x + 13 = 6 x + 1 , then a || b. If 8 y + 3 + 73 = 180 , then m || n. 5 x + 13 = 6 x + 1 8 y + 76 = 180 - 5 x - 1 = - 5 x - 1 8 y = 104 12 = x y = 13

Example #2. Given the following information, Example #2. Given the following information, name the lines (if any) that must be parallel. p || k (a) ∠1 ≅ ∠8 _________ WHY? t p k n 3 4 5 2 6 1 1 8 8 a 10 9 7

Example #2. Given the following information, Example #2. Given the following information, name the lines (if any) that must be parallel. none (b) ∠4 ≅ ∠6 _________ t p k n 3 4 4 5 2 6 6 1 8 a 10 9 7

Example #2. Given the following information, Example #2. Given the following information, name the lines (if any) that must be parallel. none (c) ∠10 ≅ ∠5 _________ t p k n 3 4 5 5 2 6 1 8 a 10 10 9 7

Example #2. Given the following information, Example #2. Given the following information, name the lines (if any) that must be parallel. p || k (d) ∠5 ≅ ∠3 _________ WHY? t p k n 3 3 4 5 5 2 6 1 8 a 10 9 7

Example #2. Given the following information, Example #2. Given the following information, name the lines (if any) that must be parallel. a || n (e) ∠2 ≅ ∠7 _________ WHY? t p k n 3 4 5 2 2 6 1 8 a 10 9 7 7

Example #2. Given the following information, Example #2. Given the following information, name the lines (if any) that must be parallel. p || k (f) m∠3 + m ∠ 4 = 180º _________ t p k WHY? n 3 3 4 4 5 2 6 1 8 a 10 9 7

Example #2. Given the following information, Example #2. Given the following information, name the lines (if any) that must be parallel. none (g) m∠5 + m ∠ 6 = 180º _________ t p k n 3 4 5 5 2 6 6 1 8 a 10 9 7

Example #2. Given the following information, Example #2. Given the following information, name the lines (if any) that must be parallel. a || n (h) m∠6 + m ∠ 9 = 180º _________ t p k WHY? n 3 4 5 2 6 6 1 8 a 9 10 9 9 7

Example #2. Given the following information, Example #2. Given the following information, name the lines (if any) that must be parallel. none (i ) m∠9 + m ∠ 10 = 180º _________ t p k WHY? n 3 4 5 2 6 1 8 a 10 10 9 9 7

How can you apply parallel lines (planes) to make deductions? Assignment Written Exercises on pages 87 & 88 RECOMMENDED: 17, 29 REQUIRED: 1 to 15 odd numbers, 19, 27, 28 Prepare for a quiz on Lessons 3-1 to 3-3: Parallel Lines and Planes How can you apply parallel lines (planes) to make deductions?