Warmup!:

Extra credit opportunity!Proving corollary from chp. 7 (3 points)

p

r z

yx

Then prove that y= z(z+r)

Prove that p= r(r+z)

Don’t need statement/reason proof, just algebra (hint use similar triangles, not pythagorean theorem)

Theorem 7-4 Side-Splitter TheoremIf a line parallel to one side of a triangle intersects the other two sides, then it divides those sides proportionally.

B

C

G

D

F

If GB // DF,

then

FCBF

DCGD

Proof of side-splitter thm

B

C

G

D

F

1.) m<C=m<C 1.) Given2.) m<CDF=m<CGB 2.) “ “3.) ∆CDF~∆CGB 3.) AA4.) x-y=z 4.) seg. Sub.5.) r-t=w 5.) “ “6.) x = r 6.) def. similar y t7.) x -1 = r -1 7. Subtr. Prop. y t

8.) x - y = r – y 8.) subs. y y y y9.) 9.)

10.) 10.)

y

t

x

r

z

w

Prove that z/y=w/t

Examples

x

1

12

3

x 6

14

14

- x

10

Corollary to Theorem 7-4

If three parallel lines intersect two transversals, then they divide the transversals proportionally.

A

B

C G

F

DIf AD // BF // CG,

then

FGDF

BCAB

Examples

5

7

x

9

9

18

x

24 - x

Theorem 7-5 Triangle Angle-Bisector TheoremIf a ray bisects an angle of a triangle,

then it divides the opposite side into segments proportional to the other two sides.

If DG bisects ÐFDE then ED

FD

GE

GF

GF

D

E

Proof: http://www.khanacademy.org/math/geometry/triangle-properties/angle_bisectors/v/angle-bisector-theorem-proof

Examples

14

21

8

x

9

10 - x x

6

Tonight’s homework: pg 400-401 1-24