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Transcript
Chapter 3.2 Angles and Parallel Lines Check.4.8 Apply properties and theorems about angles associated with parallel and perpendicular lines to solve problems. Objective: To understand the relationship of angles created by having a transveral crossing two parallel lines. Bellwork • With a partner take one of the sheets from the side table and follow the directions. Angle Postulates and Theorems Postulate/Theorem Example Corresponding Angles Postulate If two parallel lines are cut by a transversal then each pair of corresponding angles is congruent 1 5, 2 6, 3 8, 4 7 Alternate Interior Angles Theorem If two parallel lines are cut by a transversal, then each pair of alternate interior angles is congruent 4 5, 3 6 Consecutive Interior Angles Theorem If two parallel lines are cut by a transversal, then each pair of consecutive interior angles is supplementary m4 + m6 = 180 m3+ m5 = 180 Alternate Exterior Angles Theorem If two parallel lines are cut by a transversal, then each pair of alternate exterior angles is congruent 1 8, 2 7 Perpendicular Transversal Theorem In a plane, if a line is perpendicular to one of two parallel lines, then it is perpendicular to the other 1 3 5 8 7 6 2 4 Corresponding Angles Postulate If two parallel lines are cut by a transversal then each pair of corresponding angles is congruent 1 5 1 3 2 6 3 8 4 7 5 8 7 6 2 4 Determine Angle Measures 1 3 5 8 7 6 2 4 What does this Assume? The lines are parallel In the figure, m3 = 133 Find m6 • 3 8 • Corresponding Angles • 8 6 • Vertical Angles • 3 6 • Transitive Property • m3 = m6 • Definition of congruent Angles • 133 = m6 • Substitution Determine Angle Measures 10 12 13 x y 14 15 16 17 11 In the figure, x || y, m11 = 51 Find m16 • 51 Perpendicular Transversal Theorem In a plane, if a line is perpendicular to one of two parallel lines, then it is perpendicular to the other t 1 p 2 q • p||q, t p • 1 is a right angle • m1 = 90 • 1 2 Given : p||q, t p Prove : t q • m1=m2 • m2 = 90 • 2 is a right angle • tq • Given • Def of lines • Definition of right angle • Corresponding Angles Postulate • Definition of congruent angles • Substitution • Definition of Right Angles • Def of lines Find the measure of GHI • • • • • Draw an Auxiliary Line EHK AEK, alternate interior angles m EHK = 40 FHK HFC, alternate interior angles mFHK= 70 A G E B 40 J H 40 70 • GHI = 40 + 70=110 C K 70 D F I Find the measure of GHI • • • • • Draw an Auxiliary Line EHK AEK, alternate interior angles m EHK = 60 FHK HFC, alternate interior angles mFHK= 65 A G E B 60 J 60 65 • GHI = 60 + 65 = 125 65 C F I H K D 1 m1 = 3x +40, m2 = 2(y-10), and m3 = 2x + 70 Find x and y F 3 G E 2 4 H m1 = m3 By Corresponding Angles 3x +40 = 2x + 70 -2x-40 -2x -40 x = 30 m1 = m2 Alternate Exterior Angles 3x +40 = 2(y-10) 3(30) + 40 = 2y -20 130 = 2y -20 150 = 2y 75 = y Practice Assignment • Block Page 181, 12 - 36 every 4th