3-5 Proving Lines Parallel b c Postulate 3.4: If two lines in a plane are cut by a transversal so that corresponding angles are congruent, then the lines are parallel. Line a would then be parallel to line b.

Construction Using a compass and straightedge: follow the steps to draw a parallel line through a point not on a line. p. 172

Parallel Postulate Postulate 3.5: If given a line and a point not on the line, then there exists exactly one line through the point that is parallel to the given line.

Example If m1 = 5x + 10 and m2 = 6x – 4, find x so that a || b. c 2 1 If m1 = 5x + 10 and m2 = 6x – 4, find x so that a || b. They are corresponding angles so for the lines to be parallel they must be congruent. 1 2 m 1 = m 2 5x + 10 = 6x – 4 10 = x – 4 14 = x

Theorems Theorem 3.5: If two lines in a plane are cut by a transversal so that a pair of alternate exterior angles is congruent, then the two lines are parallel. Theorem 3.6: If two lines are cut by a transversal so that a pair of consecutive interior angles are supplementary, then the two lines are parallel.

More Theorems Theorem 3.7: If two lines in a plane are cut by a transversal so that a pair of alternate interior angles is congruent, then the two lines are parallel. Theorem 3.8: If two lines are perpendicular to the same line, then the two lines are parallel.

Example Determine which lines are parallel, if any. What theorems were used? 68 112 E F D B C A

Example 9x = 10x – 8 -x = -8 x = 8 Are you done? b k f 10x - 8 9x 9x = 10x – 8 -x = -8 x = 8 Are you done? No. You need to find the angle measure. 10(8) – 8 = 80 – 8 = 72 180 - 72 = 108 Find the value of x so that a || b. Then find kfb. The angles are alternate exterior angles therefore they must be congruent.

Review Example 3 Proof p. 175 Do “Check your Progress” #4 p. 175.

Practice Try page 175-176 #1-7 before you begin your homework. Homework #22 Practice Worksheet p. 176 8-18 even, 29, 37-38