Question

$\overline{tw} \cong \overline{sw}$ and $\overline{uw} \cong \overline{vw}$. complete the proof that $\overline{sv} \cong \overline{tu}$.

statement	reason
2 $\overline{uw} \cong \overline{vw}$	given
3 $\angle swv \cong \angle twu$	vertical angle theorem
4

Explanation:

Step1: Identify Congruent Triangles

We have two sides and the included angle for triangles \( \triangle SWV \) and \( \triangle TWU \). \( \overline{TW} \cong \overline{SW} \), \( \overline{UW} \cong \overline{VW} \), and \( \angle SWV \cong \angle TWU \) (vertical angles). By SAS (Side - Angle - Side) congruence criterion, \( \triangle SWV \cong \triangle TWU \).

Step2: Corresponding Parts of Congruent Triangles

Since \( \triangle SWV \cong \triangle TWU \), their corresponding sides \( \overline{SV} \) and \( \overline{TU} \) are congruent. So the statement for step 4 is \( \overline{SV} \cong \overline{TU} \) and the reason is "Corresponding Parts of Congruent Triangles are Congruent (CPCTC)" or from the congruence of triangles \( \triangle SWV \cong \triangle TWU \) by SAS, we can conclude \( \overline{SV} \cong \overline{TU} \).

Answer:

Statement 4: \( \overline{SV} \cong \overline{TU} \)
Reason 4: Corresponding Parts of Congruent Triangles are Congruent (or \( \triangle SWV \cong \triangle TWU \) by SAS, so \( \overline{SV} \cong \overline{TU} \))