Indirect MeasurementIndirect Measurementand Additionaland Additional

Similarity Theorems 8.5Similarity Theorems 8.5

• Learn the triangle angle bisector theorem.

homeworkhomework

• Learn the proportional altitudes theorem.

• Learn the proportional medians theorem.

Proportional Angle Bisectors TheoremProportional Angle Bisectors Theorem

homeworkhomework

Proportional Altitudes TheoremProportional Altitudes Theorem

homeworkhomework

Proportional Medians TheoremProportional Medians Theorem

homeworkhomework

Proportional Angle Bisector Theorem

Find PS and SR.

40(x – 2) = 32(x + 5)

40x – 80 = 32x + 160

8x = 240

x = 30

PS = x – 2 SR = x + 5

RS

RQ

PS

PQ

5x

40

2x

32

PS = 28 SR = 35

homeworkhomework

A. A

B. B

C. C

D. D

The drawing of the table below has legs AE and CG. AC measures 12 inches and GE measures 36 inches. If BD measures 7 inches, what is the measure of DF and what is the height of the table?

Proportional Altitudes Theorem

12x = 252

x = 21

DF

GE

BD

AC

x

36

7

12

Therefore, DF = 21, and the table is 28 inches tall. homeworkhomework

Proportional Medians TheoremIn the figure, EFD ~ JKI. EG is a median of EFG and JL is a median of JKL. Find JL if EF = 36, EG = 18, and JK = 56.

1008 = 36x

x = 28

JK

EF

JL

EG

56

36

x

18

Therefore, JL = 28.

3618

56

E

D G F

K

J

I L

x

homeworkhomework

Use indirect measure to find the missing value.

26d = 800

40

d

26

20

inches13

1030d

homeworkhomework

Use indirect measure to find the missing value.

10x = 360

45

x

10

8

x = 36

homeworkhomework

Use indirect measure to find the missing value.

4t = 108

18

t

4

6

x = 27 feet

homeworkhomework

homeworkhomework

Use indirect measure to find the missing value.

4x = 55

10

x

4

5.5

x = 13.75 feet

SR = 25, ST = 15

Use the Triangle Angle Bisector Theorem find ST and SR

7.5x + 37.5 = 12.5x – 62.5

–5x = –100

TV

ST

RV

RS

5.7

5x

5.12

5x

x = 20

homeworkhomework

EXAMPLE 4

In the diagram, QPR RPS. Use the given side lengths to find the length of RS .

7x = 195 – 13x

x = 9.75

SR

PS

QR

PQ

x

13

x15

7

homeworkhomework

homeworkhomework

Use the Triangle Angle Bisector Theorem find x.

42x2 = 50x2 – 20x

8x2 – 20x = 0

x6

4x10

x5

x7

4x(2x – 5) = 0

x = 0 or x = 2.5

homeworkhomework

4x = 36 – 6x

x = 3.6

x6

4

x

6

Use the Triangle Angle Bisector Theorem find x.

10x = 36

AssignmentAssignment

8.5 Indirect 8.5 Indirect MeasurmentMeasurment