Question

complete the proof that $\triangle stu \cong \triangle rvu$.

(image of the geometric figure with points r, s, v, t, u, right angles at v and t, congruent segments marked, and congruent angles at u)

statement	reason
2 $\overline{uv} \perp \overline{rv}$	given
3 $\overline{st} \cong \overline{rv}$	given
4 $\angle ruv \cong \angle sut$	given
5 $\angle v \cong \angle t$	all right angles are congruent
6 $\triangle stu \cong \triangle rvu$

Explanation:

Step1: Identify triangle congruence criteria

We have two triangles, $\triangle STU$ and $\triangle RVU$. Let's list the known congruent parts:

• From step 3, $\overline{ST} \cong \overline{RV}$ (a side).
• From step 5, $\angle V \cong \angle T$ (an angle).
• From step 4, $\angle RUV \cong \angle SUT$ (another angle). Wait, no, actually, let's re - check. Wait, $\angle V$ and $\angle T$ are right angles (from steps 1 and 2, since $ST\perp TU$ and $UV\perp RV$, so $\angle T$ and $\angle V$ are right angles, hence congruent). Also, we have $\overline{ST}\cong\overline{RV}$ (step 3), and $\angle RUV\cong\angle SUT$ (step 4). Wait, actually, let's use the AAS (Angle - Angle - Side) congruence criterion. In $\triangle STU$ and $\triangle RVU$:
• $\angle T\cong\angle V$ (right angles, step 5).
• $\angle SUT\cong\angle RUV$ (step 4).
• $\overline{ST}\cong\overline{RV}$ (step 3).

So by the AAS (Angle - Angle - Side) congruence postulate, $\triangle STU\cong\triangle RVU$.

Answer:

AAS (Angle - Angle - Side) Congruence Postulate