Triangle proportionality theorem

10

5

7

4

BE

CD

AD

ED

9

8

5

7

4

9

3

7

1

2

3

4

Line segment TU is parallel to line segment RS because  $\frac{32}{36}=\frac{40}{45}$   .

Line segment TU is not parallel to line segment RS because  $\frac{32}{36}\ne\frac{40}{45}$  

Line segment TU is parallel to line segment RS because  $\frac{32}{45}=\frac{40}{36}$   .

Line segment TU is not parallel to line segment RS because  $\frac{32}{45}\ne\frac{40}{36}$   .

△BCD is similar to △BSR.

$\frac{BR}{RD}=\frac{BS}{SC}$

If the ratio of BR to BD is $\frac{2}{3}$ , then it is possible that BS = 6 and BC = 3.

(BR)(SC) = (RD)(BS)

$\frac{BR}{RS}=\frac{BS}{SC}$