Question

$\angle tsv \cong \angle usv$ and $\overline{tu} \perp \overline{sv}$. complete the proof that $\overline{uv} \cong \overline{tv}$.

(image of a triangle with vertices s, t, u and a point v on tu)

statement	reason
2 $\overline{tu} \perp \overline{sv}$	given
3 $\angle svt \cong \angle svu$	all right angles are congruent
4 $\overline{sv} \cong \overline{sv}$	reflexive property of congruence
5 $\triangle stv \cong \triangle suv$
6 $\overline{uv} \cong \overline{tv}$	cpctc

Explanation:

Step1: Identify missing reason for Step5

We have two angles and the included side congruent: $\angle TSV \cong \angle USV$, $\overline{SV} \cong \overline{SV}$, $\angle SVT \cong \angle SVU$. This matches the ASA (Angle-Side-Angle) Congruence Postulate, which states that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, the triangles are congruent.

Step2: Confirm final step validity

Once $\triangle STV \cong \triangle SUV$, corresponding parts of congruent triangles are congruent (CPCTC), so $\overline{UV} \cong \overline{TV}$, which is already provided.

Answer:

The missing reason for Statement 5 ($\triangle STV \cong \triangle SUV$) is ASA Congruence Postulate. The completed proof is valid, with $\overline{UV} \cong \overline{TV}$ confirmed by CPCTC.