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3.2 Proving Lines Parallel The secret secret impresses impresses no no one. one. The The trick trick you you use use itit for for is is everything. everything. The -Alfred Borden Proving Lines Parallel l m 1 2 Postulate 3-2: Converse of Corresponding Angles Postulate: If two lines and a transversal form corresponding angles that are congruent, then the two lines are parallel. l || m Proving Lines Parallel We know these two lines are parallel!! If Alternate Interior Angles are congruent we can assume lines are parallel too! Proving Lines Parallel l m 1 4 2 Theorem 3-5: Converse of Alternate Interior Angles Theorem If two lines and a transversal form alternate interior angles that are congruent, then the two lines are parallel. If 1 2, then l || m. Proof of Theorem 3-5 (C of AIAT) l 3 1 m Given : 2 1 2 Prove : l || m Statements Reasons 1. 2 1 1. 2. 2. Vertical 's are 3. 2 3 3. l || m 4. 4. Proving Lines Parallel We know these two lines are parallel!! If Same-Side Interior Angles are supplementary, we can assume lines are parallel too!! Proving Lines Parallel l m 1 4 2 Theorem 3-6: Converse of Same-Side Interior Angles Theorem If two lines and a transversal form same-side interior angles that are supplementary, then the two lines are parallel. If 2 and 4 are supplement ary, then l || m. Proving Lines Parallel We know these two lines are parallel!! If Alternate Exterior Angles are congruent, we can assume lines are parallel!!! Proving Lines Parallel a 1 b 3 2 Theorem 3-7: Converse of Alternate Exterior Angles Theorem If two lines and a transversal intersects form alternate exterior angles that are congruent, then the two lines are parallel. If 1 2, then a || b. Proof of Theorem 3-7 (C of AEAT) a 1 4 Given : 2 1 b Prove : a || b Statements 2 Reasons 1. 1. 2. 1 4 2. 3. 2 4 3. a || b 4. 4. Proving Lines Parallel We know these two lines are parallel!! If Same-Side Exterior Angles are supplementary, we can assume lines are parallel!!! Proving Lines Parallel a 1 b 3 2 Theorem 3-8: Converse of Same-Side Exterior Angles Theorem If two lines and a transversal intersects form same-side exterior angles that are supplementary, then the two lines are parallel. If 1 and 3 are supplementary, then a || b. Let’s Apply What We Have Learned, K? Find the value of x for which l || m l m 40° (2x + 6)° You Try One! Find the value of x for which a || b a (7x - 8)° b 62° 3.3: Parallel and Perpendicular Lines There is no spoon. There is no spoon. -Spoon Boy and Neo Relating Parallel and Perpendicular Lines a b c Theorem 3-9: If two lines are parallel to the same line, then they are parallel to each other. a || b Relating Parallel and Perpendicular Lines t m n Theorem 3-10: In a plane, if two lines are perpendicular to the same line, then they are parallel to each other. m || n Relating Parallel and Perpendicular Lines t m 1 2 n Given : m t, n t Prove : m || n Statements Reasons 1. m t, n t 1. 2. 1 and 2 are right angles 2. 3. 1 2 3. m || n 4. 4. Relating Parallel and Perpendicular Lines n l m Theorem 3-11: In a plane, if a line is perpendicular to one of two parallel lines, then it is also perpendicular to the other. nm Relating Parallel and Perpendicular Lines n 1 l 2 Given : n l, and l || m Prove : n m Statements Reasons 1. n l, and l || m 1. 2. 1 is a right 2. 3. 3. Corresp. ‘s post. 4. n m 4. m 3.2: Proving Lines Parallel 3.3: Parallel and Perpendicular Lines HOMEWORK: p.137 #1-8, 14-22, 24-30 even, 51-52, p. 144 #26-27, Checkpoint Quiz 1 (p. 153) #1-9 Get busy busy living, living, or or get get busy busy dying. dying. Get -Red