Question

use the given information to prove that $qs = tu$.
given: $qr = ru$
$rs = tr$
prove: $qs = tu$

Explanation:

Step1: Vertical angles

$\angle QRS=\angle URT$ (Vertical angles are equal)

Step2: SAS congruence

In $\triangle QRS$ and $\triangle URT$,
$QR = RU$ (Given),
$\angle QRS=\angle URT$ (Vertical angles),
$RS = TR$ (Given).
So, $\triangle QRS\cong\triangle URT$ (SAS congruence criterion)

Step3: Corresponding parts

Since $\triangle QRS\cong\triangle URT$, then $QS = TU$ (Corresponding parts of congruent triangles are equal)

Answer:

$QS = TU$ is proved as above.