Proportionalit y 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

5 X+1 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 3rd side

AD

=

AE

then DE is parallel to BC

DB EC

Proving lines parallel

• Is ST parallel to PR?

• PS = RT • SQ TQ

P 3 S 8 7 R 2 T Q

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 3x-1 4 4

practice

• Determine whether AD, BE, and CF are parallel when x = 3

4x-1 5x-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 2x-1 Q R 5 B 3

practice

• Find the length of AD

B 3 A 4 2

practice

• Are the lines parallel ?

•

Y =9, X = 9.75,C = 12, E = 13 does