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5-3 Theorems involving parallel lines Theorems involving parallel lines Theorem 5 – 8 If two lines are parallel, then all points on one line are equidistant from the other line l A B C D m Since line l is parallel to line m; AC = BD Theorems involving parallel lines Theorems 5 – 9 If three parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal A B 3 X 5 Y C 4 Given AX BY CZ, and AB = BC we can say that XY = YZ 6 Z Theorems involving parallel lines Theorem 5 – 10 A line that contains the midpoint of one side of a triangle and is parallel to another side passes through the midpoint of a third side A M B D N C If M is the midpoint of AB and MN BC. We can draw a third line from A called D, so that AD, MN, and BC are all parallel. Now we have three parallel lines cut by a transversal. We can use theorem 5-9 to say that N is the midpoint of AC Theorems involving parallel lines Theorem 5 – 11 The segment that joins the midpoints of two sides of a triangle (1) is parallel to the third side; (2) is half as long as the third side A M B N C If N is the midpoint of AB and N is the midpoint of AC we can prove that MN is parallel to BC because there is exactly one line through M parallel to BC. By Theorem 5 – 10 that line passes through N and the midpoint AC. So MN is parallel BC Example P, Q, and R are midpoints of the sides of DEF. 12 P D a) b) E What kind of figure is DPQR What is the Perimeter of DPQR R Q 8 10 F Solution P, Q, and R are midpoints of the sides of DEF. 12 D a) b) E P What kind of figure is DPQR What is the Perimeter of DPQR Q R 8 10 F a) Since RQ DE and PQ DF, Quad DPQR is a parallelogram b) RQ = ½ DE = DP = 6 and PQ = ½ DF = DR = 4. DPQR is 6 + 4 + 6 + 4 or 20 So the perimeter of