Question

use the given information to prove that $qs = tu$.

given: $qr = ru$
$rs = tr$
prove: $qs = tu$

Explanation:

Step1: Vertical angles are equal

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

Step2: Prove triangles congruent

In $\triangle QRS$ and $\triangle URT$:
$QR = RU$ (Given)
$\angle QRS=\angle URT$ (From Step 1)
$RS = TR$ (Given)
By the Side - Angle - Side (SAS) congruence criterion, $\triangle QRS\cong\triangle URT$

Step3: Corresponding parts of congruent triangles

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

Answer:

$QS = TU$ is proved as above.