5.3 Theorems Involving Parallel Lines Geometry 5.3 Theorems Involving Parallel Lines

4 UsefulTheorems Today you will learn 4 useful theorems about parallel lines. These will help you solve problems and write proofs.

Theorem If 2 lines are parallel, then all points on one line are equidistant from the other line. A B AC = BD l m C D

Theorem If 3 parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal. If AB = BC then XY = YZ A X B Y C Z

Theorem A line that contains the midpoint of one side of a triangle and is parallel to another side passes through the midpoint of the third side. X If M is the midpoint of XY then N is the midpoint of XZ M N Y Z

Theorem A segment that joins the midpoints of 2 sides of a triangle; 1. is parallel to the third side. 2. is half as long as the third side. X MN YZ MN = ½YZ M N Y Z

Find x when PQ = 3x – 9 and QR = 2x – 2. Sample Problems: Find x when PQ = 3x – 9 and QR = 2x – 2. Find PR and PS when PQ = 5. Find RS when PS = 33. Find x when PR = x + 8 and QS = 3x – 2. 5) Find QS in problem #4. A D C B P Q R S

6) Name the points shown that MUST be midpoints of the sides of the large triangle. X Y 6 A B C Z X Y 6 A B C Z 5 3 X Y 5 A B C Z 6 4

Find YZ if MN = 8. Find MN if YZ = 30. Find YZ if MN = 2k + 3. 10) Find MN if YZ = k. X Y Z M N

11) Find x, MN and YZ if MN = x and YZ = 3x – 25.

13) Find x and y if AB = 19, BC = x + 3y, and CD = 6x – y. P Q R S

Homework P. 180 #1-17

Let’s Try A Few Together Practice 18 # 6 Test 18 #2, 11, 12