Proportionality Theorems Theorem 60 1 Triangle Proportionality theorem

Proportionality Theorems

Theorem 60 -1 Triangle Proportionality theorem • If a line parallel to one side of a triangle intersects the other 2 sides, it divides those sides proportionally • AD= AE • DB EC

Using Proportionality to find unknowns • Find the length of line segment RT • QS = RT • SP TP

Find x X+1 5 X+3 10

Theorem 60 -2 Converse of Triangle Proportionality theorem • If a line divides 2 sides of a triangle proportionally, then it is parallel to the 3 rd side • AD= AE then DE is parallel to BC • DB EC

Proving lines parallel • Is ST parallel to PR? • PS = RT P 3 • SQ TQ S 8 Q 7 R 2 T

Theorem 60 -3 • If parallel lines intersect transversals, then they divide the transversals proportionally • PQ =JK • QR KL

• If parallel lines divide a transversal into congruent segments, then the segments are in a 1: 1 ratio. By theorem 60 -3, any other transversal cut by the same parallel lines will be divided into segments that also have a 1: 1 ratio, so they will also be congruent. 3 3

Theorem 60 -4 • If parallel lines cut congruent segments on one transversal, then they cut congruent segments on all transversals. • IF AB = BC, then DE = EF

Finding segment lengths with intersecting transversals • Find the length of AB X+3 3 x-1 4 4

practice • Determine whether AD, BE, and CF are parallel when x = 3 4 x-1 5 x-3 14 14 x y z

Find the length of EB • ED is parallel to BC B E 4 A 6 D 10 C

Find the length of PQ • PR is parallel to AB P x A 2 x-1 Q 5 B 3 R

practice • Find the length of AD 3 B A 4 2

practice • Are the lines parallel ? • Y =9, X = 9. 75, C = 12, E = 13 does