Geometry Notes G.2 G.3 Parallel/Perpendicular Lines, Slope Mrs. Grieser Name: __________________________________________ Date: ________________ Block: _______ Proving Lines are Parallel Review: Given p q, what is the converse?___________ The converses of the postulates and theorems from the last lesson are true: Corresponding Angles Converse Postulate If two lines are cut by a transversal such that the corresponding angles are congruent, then the lines are parallel. Alternate Interior Angles Converse Theorem If two lines are cut by a transversal such that the alternate interior angles are congruent, then the lines are parallel. Alternate Exterior Angles Converse Theorem If two lines are cut by a transversal such that the alternate exterior angles are congruent, then the lines are parallel. Consecutive Interior Angles Converse Theorem If two lines are cut by a transversal such that the consecutive interior angles are supplementary, then the lines are parallel. Examples: a) Find the value of x that b) Can we prove m || n? c) Can we prove a || b? makes m || n. Explain. Explain. Transitive Theorem of Parallel Lines If two lines parallel to the same line, then they are parallel to each other. Example: In the figure at right, each rung of the ladder is parallel to the rung directly beneath it. Explain why the top rung is parallel to the bottom rung. Using a paragraph proof: Rung 1 is parallel to _______, rung 2 is parallel to ______; rung 1 is parallel to rung 3. We can continue with this reasoning to show that the top rung is parallel to the bottom rung. Geometry Notes G.2 G.3 Parallel/Perpendicular Lines, Slope Mrs. Grieser Page 2 Algebraically Proving Lines Parallel/Perpendicular change in y Δy y y Blast from the past! Slope of a line = m = 2 1 change in x Δx x2 x1 positive slope: as x increases, y _________ (line ____) negative slope: as x increases, y _________ (line ____) 0 slope: line is __________ (line ____) undefined slope: line is ___________ (line ____) Slopes of Parallel Lines Postulate In a coordinate plane, two nonvertical lines are parallel IFF they have the same slope. Any two vertical lines are parallel. Slopes of Perpendicular Lines Postulate In a coordinate plane, two nonvertical lines are perpendicular IFF the product of their slopes is -1. Horizontal lines are perpendicular to vertical lines. ** Perpendicular lines have slopes that are negative reciprocals of each other (same as product of slopes = -1). ** Example: Which lines, if any, are parallel? mk1 = _____ mk2 = _____ mk3 = _____ Conclusion:__________________________ Examples: a) Graph the line parallel to line AB that passes b) Graph the line perpendicular to AB that through point P. passes through point P. mAB =________ mAB =________ m|| line = _______ m line = ______ You try: Tell whether the lines through the given points are parallel, perpendicular, or neither… a) line 1: (-5, 2), (-3, 5) b) line 1: (-2, 3), (-5, 2) c) line 1: (-3, -2), (1, 2) line 2: (-2, 2), (1, 0) line 2: (4, 1), (5, 3) line 2: (1, 3), (4, 6) .