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## Geometry

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**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 N M 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 N M Y Z**A**P B Q C R D S 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.**X**X X 6 4 3 A A B A B B 4 5 3 Y Y Z Z Y Z C C C 5 6 6 6 6) Name the points shown that MUST be midpoints of the sides of the large triangle.**X**N M Z Y • Find YZ if MN = 8. • Find MN if YZ = 30. • Find YZ if MN = 2k + 3. • 10) Find MN if YZ = k.**X**N M Z Y 11) Find x, MN and YZ if MN = x and YZ = 3x – 25. 12) Find if and .**A**P B Q C R D S 13) Find x and y if AB = 19, BC = x + 3y, and CD = 6x – y.**Homework**P. 180 #1-17**Let’s Try A Few Together**• Practice 18 # 6 • Test 18 #2, 11, 12