Theorems Involving Parallel Lines and Triangles

1. Theorems Involving Parallel Lines and Triangles Keystone Geometry: Using Theorems, Midpoints, and Triangles

2. Theorem: • If three parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal. • If and AC = CE , then BD = DF.

3. 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. • If M is the midpoint of XY and , then N is the midpoint of XZ.

4. The Midsegment • The segment that joins the midpoints of two sides of a triangle is called the Midsegment. • The Midsegment: • Is parallel to the third side. • Is half as long as the third side. • If M is the midpoint of XY and N is the midpoint of XZ, then MN || YZ and MN = 1/2 YZ.

5. are parallel, with AB = BC = CD. • If QR = 3x and RS = x+12, then x=____ • If PQ = 3x - 9 and QR = 2x - 2, then x=____ • If QR = 20, then QS=____ • If PQ = 6, then PR=____ and PS=____ • If PS = 21, then RS=____ • If PR = 7x - 2 and RS = 3x + 4, then x=____ • If PR = x + 8 and QS = 3x - 2, then x=____ A P 6 Q B C R 7 D S 40 12 18 7 10 5

6. Example: M is the midpoint of XY and N is the midpoint of XZ. 12 If MN = 6, then YZ =_____. If YZ = 20, then MN =_____. If MN = x and YZ = 3x - 25, Then x =_____, MN = _____, And YZ =_____. If <XMN = 40, then <XYZ =_____. 10 25 25 50 40 They are Corresponding Angles.