Question

use the converse of the side-splitter theorem to determine if tu || rs. which statement is true?
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} \
eq \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} \
eq \frac{40}{36}\\).

Explanation:

Step1: Recall converse of side-splitter theorem

If a line divides two sides of a triangle proportionally, it is parallel to the third side. Here, TU divides QS and QR into segments QT=32, TR=36, QU=40, US=45. Check if $\frac{QT}{TR} = \frac{QU}{US}$.

Step2: Calculate the ratios

$\frac{QT}{TR} = \frac{32}{36} = \frac{8}{9}$; $\frac{QU}{US} = \frac{40}{45} = \frac{8}{9}$. The ratios are equal.

Answer:

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