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Theorems Involving Parallel Lines and Triangles

This overview discusses key theorems involving parallel lines and triangles, focusing on midpoints and the properties of midsegments. Key theorems include the congruence of segments cut off by parallel lines on different transversals and the characteristics of the line connecting midpoints of triangle sides. The midsegment is shown to be parallel to the third side and half its length, providing insights into triangle geometry and problem-solving strategies. Examples illustrate these concepts through practical applications, enhancing comprehension of theorems related to triangle geometry.

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Theorems Involving Parallel Lines and Triangles

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  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.

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