Three Theorems Three Theorems Involving ProportionsInvolving Proportions

Section 8.5Section 8.5

A T

F

B N

Given: AT ║ BN

FA = 12, FT = 15

AT = 14, AB = 8

Find: length BN & NT

Prove similar triangles firstProve similar triangles first

base

side

14

12

BN

20=

14

15

31

23

15 TN=

BN = 23 BN = 23 11//33

TN = 10TN = 10

Theorem 65: If a line is ║to one side of a Δ and intersects the other 2 sides, it divides those 2 sides proportionally. (Side splitter theorem)

A B

C

E

D

FG

Given: AB CD EF

Conclusion:

AC

CE

BD

DF=

Draw AF

In Δ EAF, CEAC

= GF

AG

Bottom

Top

In Δ ABF,

GF

AG= DF

BD

Substitute BD for AG and DF for GF,

substitution or transitive

CE

AC= DF

BD

Theorem 66: If three or more parallel lines are intersected by two transversals, the parallel lines divide the transversals proportionally.

DC

B

AGiven: ∆ ABD

AC bisects <BAD

Prove:

BC

CD

AB

AD=

Set up ProportionsSet up Proportions

AB

BC

AD

CD= Rightside

Rightbase

leftside

Leftbase =

BC • AD = AB • CD Means Extreme

CD

BC

Therefore:

=AD

ABRightside

Leftside

Rightbase

Leftbase=

Theorem 67: If a ray bisects an angle of a triangle, it divides the opposite side into segments that are proportional to the adjacent sides. (Angle Bisector Theorem)

a

G

ZS

Y

I

P

X

W

b

c

d

2

3

4

Given: a, b, c and d are parallel lines, Lengths given, WZ = 15

Find: WX, XY and YZ

According to Theorem 66, the ratio of WX:XY:YZ is 2:3:4.Therefore, let WX = 2x, XY = 3x and YZ = 4x2x + 3x + 4x = 159x = 15x = 5/3

WX = 10/3 = 3 1/3 XY = 5YZ = 20/3 = 6 2/3