7-4 Parallel Lines and Proportional Parts Geometry

Use proportional parts with triangles. Use proportional parts with parallel lines. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

Theorems 7.5 Triangle Proportionality Theorem If a line parallel to one side of a triangle intersects the other two sides, then it divides the two side proportionally. If TU ║ QS, then RT RU = TQ US

Ex. 1 In ΔPQR, ST//RQ. If PT = 7.5, TQ = 3, and SR = 2.5, find PS.

Ex. 2 AB is parallel to MN, find x. 10 8 A B x 5 N M

Ex. 3 NP is parallel to RS, find x. M 10 12 N P 15 x R S

Ex. 4: Finding the length of a segment In the diagram AB ║ ED, BD = 8, DC = 4, and AE = 12. What is the length of EC?

Theorems 7.6 Converse of the Triangle Proportionality Theorem If a line divides two sides of a triangle proportionally, then it is parallel to the third side. RT RU If , then TU ║ QS. = TQ US

Ex. 5: Determining Parallels Given the diagram, determine whether MN ║ GH. LM 56 8 = = MG 21 3 LN 48 3 = = NH 16 1 8 3 ≠ 3 1 MN is not parallel to GH.

Theorem 7.7 A midsegment of a triangle is parallel to one side of the triangle, and its length is one half the length of that side. AB = ½ PS A B P S

Proportional Parts of Parallel Lines If three parallel lines intersect two transversals, then they divide the transversals proportionally. If r ║ s and s║ t and l and m intersect, r, s, and t, then UW VX = WY XZ

Ex. 3: Using Proportionality Theorems In the diagram 1  2  3, and PQ = 9, QR = 15, and ST = 11. What is the length of TU?

9 ● TU = 15 ● 11 Cross Product property PQ ST Parallel lines divide transversals proportionally. = QR TU 9 11 = Substitute 15 TU 9 ● TU = 15 ● 11 Cross Product property 15(11) 55 TU = = Divide each side by 9 and simplify. 9 3 So, the length of TU is 55/3 or 18 1/3.

Class work page 495 Problems 1-9 Homework on page 496 Problems 10-17