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Geometry 6.26.3 Parallel Lines January 05, 2017 • Ch 5 Review Problems pp. 206209 #1550 due FRIDAY 01/06 • TEST #2 Monday 01/09?? • Ch 6 Review Problems pp. 250254 #919, 3353 6.2 – Proving Lines Parallel Def: Two lines are parallel iff they lie in the same plane and do not intersect. A transversal is a line that intersects two or more lines in different points. l1 l2 t 1 Geometry 6.26.3 Parallel Lines January 05, 2017 When a transversal intersects two lines that lie in the same plane, it forms pairs of angles that are given special names: Alternate interior angles Corresponding angles Interior angles on the same side of the transversal Theorem 17: Equal corresponding angles mean that lines are parallel. 2 Geometry 6.26.3 Parallel Lines January 05, 2017 Corollary 1: Equal alternate interior angles mean that lines are parallel. Corollary 2: Supplementary interior angles on the same side of a transversal mean that lines are parallel. Corollary 3: In a plane, two lines perpendicular to a third line are parallel. Corollary 1: Equal alternate interior angles mean that lines are parallel. Given: 1= 2 Prove: a||b 1 2 Proof: Statements 3 Reasons a b c 1. Given 2. Vertical angles are equal 3. Substitution 4. Equal corresponding angles mean that lines are parallel. 3 Geometry 6.26.3 Parallel Lines January 05, 2017 Corollary 2: Supplementary interior angles on the same side of a transversal mean that lines are parallel. Given: 1 and 2 are supplementary Prove: a||b a 1 2 Proof: Statements Reasons 1. Given 2. The angles in a linear pair are supplementary 3. Supplements of the same angle are equal 4. Equal corresponding angles mean that lines are parallel. b 3 c Corollary 3: In a plane, two lines perpendicular to a third line are parallel. Given: a c and b c 1 Prove: a||b 2 Proof: Statements Reasons 1. Given 2. Perpendicular lines form right angles 3. All right angles are equal 4. Equal corresponding angles mean that lines are parallel. a b c 4 Geometry 6.26.3 Parallel Lines January 05, 2017 Given: ABD and BDE are right angles A B Prove:AB||DE 29. Proof: Statements Reasons 1. ABD and BDE are right angles Given C D 2. AB BD and BD DE E 3. AB||DE 30. Proof: Statements Reasons 1. ABD and BDE are right angles Given 2. ABD= BDE 3. AB||DE 39. E Given: AE=AD and E= BCE D C Prove: AD||BC A B 5 Geometry 6.26.3 Parallel Lines January 05, 2017 6.3 – The Parallel Postulate Construction 7: To construct a line parallel to a given line through a given point. 183 122° Theorem 17: Equal corresponding angles mean that lines are parallel. Corollary 1: Equal alternate interior angles mean that lines are parallel. Corollary 2: Supplementary interior angles on the same side of a transversal mean that lines are parallel. Corollary 3: In a plane, two lines perpendicular to a third line are parallel. The Parallel Postulate – Through a point not on a line, there is exactly one line parallel to the given line. Theorem 18: In a plane, two lines parallel to a third line are parallel to each other. 6