Question

u is the midpoint of \\(\overline{qs}\\). complete the proof that \\(\triangle rsu \cong \triangle tqu\\).

1. u is the midpoint of \\(\overline{qs}\\) - given
2. \\(\overline{qs} \perp \overline{qt}\\) - given
3. \\(\overline{qs} \perp \overline{rs}\\) - given
4. \\(\angle q \cong \angle s\\) - all right angles are congruent
5. \\(\overline{qu} \cong \overline{su}\\) - definition of midpoint
6. \\(\angle qut \cong \angle rus\\) - vertical angle theorem
7. \\(\triangle rsu \cong \triangle tqu\\) - reason to be filled

Explanation:

Step1: Identify congruent angles

$\angle Q \cong \angle S$ (from statement 4)

Step2: Identify congruent side

$\overline{QU} \cong \overline{SU}$ (from statement 5)

Step3: Identify congruent angles

$\angle QUT \cong \angle RUS$ (from statement 6)

Step4: Apply congruence postulate

For $\triangle RSU$ and $\triangle TQU$, we have two pairs of congruent angles and the included congruent side, so $\triangle RSU \cong \triangle TQU$ by ASA Congruence Postulate.

Answer:

ASA (Angle-Side-Angle) Congruence Postulate