Question

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

(there is a diagram of triangle with vertices t, s, r, and u on st with a right angle at u between u and r. then a table with statements and reasons:

1. u is the midpoint of \\(\overline{st}\\) - given
2. \\(\overline{st} \perp \overline{ru}\\) - given
3. \\(\angle rus \cong \angle rut\\) - all right angles are congruent
4. \\(\overline{su} \cong \overline{tu}\\) - definition of midpoint
5. blank for statement )

Explanation:

Step1: Identify congruent sides/angles

1. $U$ is midpoint of $\overline{ST}$ (Given)
2. $\overline{ST} \perp \overline{RU}$ (Given)
3. $\angle RUS \cong \angle RUT$ (Right angles ≅)
4. $\overline{SU} \cong \overline{TU}$ (Definition of midpoint)
5. $\overline{RU} \cong \overline{RU}$ (Reflexive property)

Step2: Apply SAS congruence

$\triangle RTU \cong \triangle RSU$ by SAS (Side-Angle-Side) congruence postulate: two sides and the included angle of $\triangle RTU$ are congruent to the corresponding two sides and included angle of $\triangle RSU$.

Answer:

$\triangle RTU \cong \triangle RSU$