3-2 Angles and Parallel Lines 1 2 3 4 a b k Postulate 3.1 Corresponding Angles If two parallel lines are cut by a transversal, then each pair of corresponding angles is congruent. There will be four sets of corresponding angles each time two lines are cut. Refer to the figure above. In the figure a || b, and k is the transversal. Which angle is congruent to 1? Explain your answer. 3, because it is corresponding.
Example m 3 + m 4 = 180 m 1 + m 4 = 180 m 1 + 60 = 180 m 1 = 120 Try “Check Your Progress” on page 149, #1 Find the measure of 1, if 4 = 60. 1 corresponds to 3 3 and 4 are a linear pair = 180
Theorems 1. 3 6 and 4 5 2. 3 + 5 = 180 and 4 + 6 = 180 7 8 6 5 3 4 Theorem 3.1: Alternate Interior Angles – if two parallel lines are cut by a transversal, then each pair of alternate interior angles is congruent. Theorem 3.2: Consecutive Interior Angles: If two parallel lines are cut by a transversal, then each pair of consecutive interior angles is supplementary. Theorem 3.3: Alternate Exterior Angles: If two parallel lines are cut by a transversal, then each pair of alternate exterior angles is congruent. 1. 3 6 and 4 5 2. 3 + 5 = 180 and 4 + 6 = 180 3. 1 8 and 2 7
Example In the figure above, a || b and k is a transversal. Find m 2. 1 2 a b k (4x – 5) (3x + 10) 1 + 2 = 180 4x – 5 + 3x + 10 = 180 7x + 5 = 180 7x = 175 x = 25 Are we done? m 1 = 4(25) – 5 100 – 5 95 m 2 = 3(25) + 10 75 + 10 85 In the figure above, a || b and k is a transversal. Find m 2. 1 4x – 5 and 2 3x + 10 1 and 2 are consecutive interior angles so they are supplementary
Theorem 3.4: Perpendicular Transversal In a plane, if a line is perpendicular to one of two parallel lines, then it is perpendicular to the other. Example: Use Example #3 (pg. 151) to answer “Check Your Progress” #3. m2 = 4x + 7 & m3 = 5x –13 4x + 7 = 5x – 13 7 = x – 13 20 = x m 3 = 5x – 13, so m 3 = 5(20) – 13 m 3 = 100 – 13 m 3 = 87
Proof Take some time to take look at the proof shown on page 150. Proving Theorem 3.4 Practice Problems: Do #1-6 on page 152 before you begin your homework.