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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 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 If two parallel lines are cut by a transversal, then corresponding angles are congruent. || - lines ⇒ corr. ∠’s 1 x 4 5 y 8 3 6 7 2 Theorem 3-2 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 3 k 1 n 2 t Given: k || n transversal t cuts k and n 3 k 1 Prove: ∠1 ∠ 2 n 2 Proof: Statements See page 78! k || n 1. ___________________________________ 2. ___________________________________ 3. ___________________________________ 4. ___________________________________ Reasons Given _____________________________________________ Vert. ∠’s R ∠1 ∠ 3 _____________________________________________ ∠3 ∠ 2 _____________________________________________ ∠1 ∠ 2 _____________________________________________ || - lines ⇒ Corr. ∠’s Transitive Property Theorem 3-3 If two parallel lines are cut by a transversal, then same side interior angles are supplementary. || - lines ⇒ SSIA Supp. t Given: k || n transversal t cuts k and n Prove: ∠1 is supplementary to ∠ 4 k 1 n 4 2 t Given: k || n transversal t cuts k & n Prove: ∠1 is suppl. to ∠ 4 k 1 n 4 2 Proof: Statements Reasons Given k || n _____________________________________________ ∠1 ∠ 2 _____________________________________________ 1. ____________________________________ 2. ____________________________________ 3. ____________________________________ _____________________________________________ 4. ____________________________________ m∠2 + m ∠ 4 = 180º _____________________________________________ 5. ____________________________________ _____________________________________________ 6. ∠1 is supplementary to ∠ 4 ____________________________________ _____________________________________________ m∠1 = m ∠ 2 m∠1 +m ∠ || - lines ⇒ AIA Def. of ∠’s Angle Addition Postulate Substitution Prop. Def. of Supp. ∠’s 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 l 1 n 2 t ⊥ l ; l || n Prove: t⊥n t This proof is on page 81 #13. Given: transversal t cuts l and n t ⊥ l ; l || n Prove: t⊥n Proof: 2. 1 n 2 t Statements 1. l t⊥l m ∠ 1 = 90º l || n 3. ________________________ Reasons Given _____________________________________________ Def. of ⊥ lines _____________________________________________ Given || - lines ⇒ Corr. ∠’s ∠1 ∠2 or m∠1 = m∠2 _____________________________________________ 5. ________________________ m∠2 = 90º Substitution Property 4. 6. t⊥n Def. of ⊥ lines _____________________________________________ If two parallel lines are cut by a transversal, then … 1) corresponding angles are congruent . 2) alternate interior angles are congruent . 3) same-side interior angles are supplementary . Example #1. If m ∠1 = 120º, x || y, and m || n, then find all the other measures. 120º m ∠2 = ______ x 120º 6 120º 4 5 120º y 1 120º m 120º 2 3 60º n 60º m ∠3 = ______ 120º m ∠4 = ______ 120º m ∠5 = ______ 120º m ∠6 = ______ Example #2. Find the values of x, y, and z. 105º 75º (3x)º (5y)º 35 x = ______ 105º 75º zº 15 y = ______ x || y x 3 x = 105 x = 35 WHY? y 5 y + 105 = 180 5 y = 75 WHY? y = 15 75 z = ______ z = 75 WHY? Example #3. Find the values of x, y, and z. (4x)º 13 x = ______ 32º 52º 96º zº 4 x = 52 x = 13 WHY? 32º (3y +8)º 3 y + 8 = 32 3 y = 24 WHY? y=8 52º 8 y = ______ 96 z = ______ z + 32 + 52 = 180 z + 84 = 180 z = 96 WHY? Assignment Written Exercises on pages 80 to 82 RECOMMENDED: 1, 3, 5, 19 REQUIRED: 7, 9, 11, 15, 17, 21, 23, 25 Prepare for a quiz on Lesson 3-2: Properties of Parallel Lines How can you apply parallel lines (planes) to make deductions?