1. 1. Give the postulate or theorem that justifies why the triangles are similar.
ANSWER
AA Similarity Postulate
2. Solve = 16 32
12 – x x
ANSWER 8

2. Use ratios and proportions to solve geometric problems.
Target
Use ratios and proportions to solve geometric problems.
You will…
Use proportions with a triangle or parallel lines.

3. Vocabulary
Triangle Proportionality Theorem 6.4 –
If a line parallel to one side of a triangle intersects the other two sides, then it divides the two sides proportionally.
Know… so you can say QS is parallel to UT RQ QU RS ST =
Triangle Proportionality Converse Theorem 6.5 –
If a line divides two sides of a triangle proportionally, then it is parallel to the third side.

4. Vocabulary
Parallel Lines Proportionality Theorem 6.6 –
If three parallel lines intersect two transversals, then they divide the transversals proportionally. VX XZ UW WY =

5. part 1 side 1 part 2 part 1 part 2 side 1 side 2 side 2
Vocabulary
Angle Bisector Proportionality 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.
part 1
Above the bisector AD DB CA CB =
side 1
part 2
below the bisector
part 1
part 2
side 1
side 2 =
side 2

6. Find the length of a segment
EXAMPLE 1
Find the length of a segment
In the diagram, QS || UT , RS = 4, ST = 6, and QU = 9. What is the length of RQ ?
SOLUTION RQ QU RS ST =
Triangle Proportionality Theorem RQ 9 4 6 =
Substitute.
RQ = 6
Multiply each side by 9 and simplify.

7. EXAMPLE 2
Solve a real-world problem
Shoerack
On the shoerack shown, AB = 33 cm, BC = 27 cm, CD = 44 cm,and DE = 25 cm, Explain why the gray shelf is not parallel to the floor.
SOLUTION
Find and simplify the ratios of lengths determined by the shoerack. CD DE 44 25 = CB BA 27 33 = 9 11
Because , BD is not parallel to AE .
So, the shelf is not parallel to the floor. 44 25 9 11 =

8. GUIDED PRACTICE
for Examples 1 and 2
1. Find the length of YZ .
315 11
ANSWER
2. Determine whether PS || QR .
ANSWER
parallel

9. EXAMPLE 3
Use Theorem 6.6
City Travel
In the diagram, 1, 2, and are all congruent and GF = 120 yards, DE = 150 yards, and CD = 300 yards.
Find the distance HF between Main Street and South Main Street. FG GH ED DC =
SOLUTION
Corresponding angles are congruent, so FE, GD , and HC are parallel. Use Theorem 6.6.

10. The distance between Main Street and South Main Street is 360 yards.
EXAMPLE 3
Use Theorem 6.6 HG GF CD DE =
Parallel lines divide transversals proportionally.
HG + GF GF
CD + DE DE =
Property of
proportions. HF
120
150 =
Substitute.
450
150 HF
120 =
Simplify. FG GH ED DC =
Multiply each
side by 120 and simplify.
HF =
The distance between Main Street and South Main Street is 360 yards.

11. EXAMPLE 4
Use Theorem 6.7
In the diagram, QPR RPS. Use the given side lengths to find the length of RS . ~
SOLUTION
Because PR is an angle bisector of QPS, you can apply Theorem 6.7. Let RS = x. Then RQ = 15 – x. RQ RS PQ PS =
Angle bisector divides opposite side proportionally.
15 – x x = 7 13
Substitute.
7x = 195 – 13x
Cross Products Property
x = 9.75
Solve for x.

12. GUIDED PRACTICE
for Examples 3 and 4
Find the length of AB. 3. 4.
ANSWER
19.2
ANSWER 2 4