Warm-Up 1In the diagram, DE is

parallel to AC. Name a pair of similar

triangles and explain why they are similar.

E

A

B

C

D

E

B

D

A

B

C

DBE ~ ABCAA (corresponding angles are congruent when parallel lines are cut by a transversal)

Warm-Up 2In the diagram, notice

that AC divides the sides of the PBD proportionally. In other words, .

What relationship exists between AC and BD? Are they parallel?

6

9

12

18

A

CP D

B

CDPC

ABPA

Warm-Up 3In the diagram, lines

AD, BE, and CF are parallel. What relationship exists between AB, DE, BC, and EF?

E

A

D

C

B

F

Warm-Up 4Ray AD is an angle

bisector. Notice that it divides the third side of the triangle into two parts. Are those parts congruent? Or is there some other relationship between them?

D

A

B

C

6.6: Use Proportionality Theorems

Objectives:1. To discover, present, and use various

theorems involving proportions with parallel lines and triangles

Proportionality Theorems!Triangle

Proportionality Theorem

If a line parallel to one side of a triangle intersects the other two sides, then it divides the two sides proportionally.

Example 1Find the length of YZ.

28.64

Proportionality Theorems!Converse of the

Triangle Proportionality Theorem

If a line divides two sides of a triangle proportionally, then it is parallel to the third side.

Example 3Determine whether PS || QR.

Yes

Example 4Find the value of x so that

.|| EDBC

x=30

Proportionality Theorems!If three parallel lines

intersect two transversals, then they divide the transversals proportionally.

Click here to investigate these realtionships

Example 5Find the length of AB.

Example 6Find the value of x.

12.5

Proportionality Theorems!Angle Bisector

Proportionality Theorem

If a ray bisects an angle of a triangle, then it divides the opposite side into segments whose lengths are proportional to the other two sides.

Example 7Find the value of x.

x=10

Example 8Find the value of x.

9.75