section 6 – 6 use proportionality theorem. theorems triangle proportionality theorem – if a line...

Section 6 – 6

Use Proportionality Theorem

TheoremsTriangle Proportionality Theorem

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

Converse of the Triangle Proportionality Theorem –

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

TheoremsTheorem 6.6 – If three parallel lines

intersect two transversals, then they divide the transversals proportionally.

Theorem 6.7 – If a ray bisects an angle of a triangle, then

it divides the opposite side into segments whose lengths are proportional to the lengths of the other two sides.

Example 1In the diagram, RS || PN, MS = 15, SN = 20, and RP = 12. What is the length of MR?

MS = MR SN RP

MR 12

Triangle Proportionality Theorem

15 = 20

Cross Multiply

20MR = 180

S

N

M

P

12

15

R

20 20

20

MR = 9

Example 2In the diagram, ABD = CBD. Use the given side lengths to find the length of DC.

Because BD is an angle bisector of ABC, we can apply Theorem 6.6.

DA = DC

BABCD

3224

40

x C

B

A

Angle bisector divides opposite side proportionally.

Substitute 40 – x = x

24 32

24x =

32(40 – x)24x = 1280 – 32x

~

56x = 1280 x = 22.9

HomeworkSection 6-6

Page 400 – 403 3 – 6, 8 – 11,

13 – 16, 30 – 33