Lesson 3-2 Properties of Parallel Lines (page 78) Essential Question How can you apply parallel lines (planes) to make deductions?

Properties of Parallel Lines Refer to the results from exercises #18, #19, and #20 on page 76 (and the results from the top of page 78). We have a choice of what to accept without proof and make a postulate. This textbook uses #18 as a postulate. This is not the same in all geometry books.

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

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

Given k || n ∠1 ≅ ∠ 3 Vert. ∠’s R ≅ ∠3 ≅ ∠ 2 || - lines ⇒ Corr. ∠’s ≅ transversal t cuts k and n Prove: ∠1 ≅ ∠2 Proof: Statements Reasons ___________________________________ _____________________________________________ t k 3 1 n 2 See page 78! k || n Given ∠1 ≅ ∠ 3 Vert. ∠’s R ≅ ∠3 ≅ ∠ 2 || - lines ⇒ Corr. ∠’s ≅ ∠1 ≅ ∠ 2 Transitive Property

Theorem 3-3 || - lines ⇒ SSIA Supp. If two parallel lines are cut by a transversal, then same side interior angles are supplementary. || - lines ⇒ SSIA Supp. Given: k || n transversal t cuts k and n Prove: ∠1 is supplementary to∠ 4 t k 1 n 4 2

k || n Given || - lines ⇒ AIA ≅ ∠1 ≅ ∠ 2 Def. of ≅ ∠’s t Given: k || n transversal t cuts k and n Prove: ∠1 is supplementary to ∠ 4 Proof: k 1 n 4 2 Statements Reasons ____________________________________ _____________________________________________ k || n Given ∠1 ≅ ∠ 2 || - lines ⇒ AIA ≅ m∠1 = m ∠2 Def. of ≅ ∠’s m∠2 + m ∠4 = 180º Angle Addition Postulate m∠1 + m ∠ 4 = 180º Substitution Prop. Def. of Supp. ∠’s ∠1 is supplementary to ∠ 4

This proof is part of your assignment. Theorem 3-4 If a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other one also. Given: transversal t cuts l & n t ⊥ l ; l || n Prove: t ⊥ n l 1 n 2 t This proof is part of your assignment. See page 81 #13 for help.

l || n m∠2 = 90º Given Def. of ⊥ lines Def. of ⊥ lines Given: transversal t cuts l and n t ⊥ l ; l || n Prove: t ⊥ n Proof: 1 l n 2 t Statements Reasons t ⊥ l _____________________________________________ m ∠ 1 = 90º _____________________________________________ _____________ __________ Given ∠1 ≅ ∠2 or m∠1 = m∠2 _____________________________________________ ________________________ Substitution Property t ⊥ n _____________________________________________ Given Def. of ⊥ lines l || n || - lines ⇒ Corr. ∠’s ≅ m∠2 = 90º Def. of ⊥ lines

If two parallel lines are cut by a transversal, then … corresponding angles are congruent . alternate interior angles same-side interior angles are supplementary .

Example #1. If m ∠1 = 120º, x || y, and m || n, then find all the other measures. WHY? 120º m ∠2 = ______ m ∠3 = ______ m ∠4 = ______ m ∠5 = ______ m ∠6 = ______ x 120º 120º 60º 6 4 5 120º 120º y 120º 2 120º 1 3 60º 120º 120º m n

Example #2. Find the values of x, y, and z. 105º 75º 35 x = ______ y = ______ z = ______ (3x)º (5y)º 105º z º 15 75º x || y 75 x y 3 x = 105 5 y + 105 = 180 z = 75 WHY? x = 35 WHY? 5 y = 75 WHY? y = 15

Example #3. Find the values of x, y, and z. 13 32º x = ______ y = ______ z = ______ (4x)º 52º 8 52º 96º 32º 96 z º (3y +8)º 4 x = 52 3 y + 8 = 32 z + 32 + 52 = 180 WHY? WHY? WHY? x = 13 3 y = 24 z + 84 = 180 y = 8 z = 96

How can you apply parallel lines (planes) to make deductions? Assignment Written Exercises on pages 80 to 82 DO NOW: 1 to 5 odd numbers GRADED: 7, 9, 11, 15, & 17 NOTE: #13 was completed in your notes. Prepare for a quiz on Lesson 3-2: Properties of Parallel Lines How can you apply parallel lines (planes) to make deductions?