Question

△tuv ≅ △tsw. complete the proof that ∠uwv ≅ ∠swv.

statement	reason
2 \\(\overline{sw} \cong \overline{uv}\\)	cpctc
3 \\(\overline{st} \cong \overline{tu}\\)	cpctc
4 \\(\overline{tw} \cong \overline{tv}\\)	cpctc
5 sv = st + tv	additive property of length
6 uw = tu + tw	additive property of length
7 sv = tu + tw	substitution
8 sv = uw	transitive property of equality
9 \\(\overline{vw} \cong \overline{vw}\\)	reflexive property of congruence
10

Explanation:

Step1: Identify Goal

We need to prove \( \angle UWW \cong \angle SWW \) (assuming a typo, likely \( \angle UWV \cong \angle SWV \) or \( \angle UWW \) is a typo for \( \angle UWV \)) using the given congruent triangles \( \triangle TUV \cong \triangle TSW \) and the steps provided.

Step2: Recall CPCTC

CPCTC (Corresponding Parts of Congruent Triangles are Congruent) is key here. After establishing side and angle congruences from the given triangle congruence, we can use CPCTC for the final angle congruence.

Step3: Analyze Step 10

We have established congruent sides and used properties of equality. Now, to prove \( \angle UWW \cong \angle SWW \) (correcting to \( \angle UWV \cong \angle SWV \) or similar), we use CPCTC because \( \triangle TUV \cong \triangle TSW \) implies corresponding angles (like \( \angle UWV \) and \( \angle SWV \)) are congruent. Wait, but looking at the table, step 10's reason should be CPCTC or another property. Wait, actually, after proving \( \triangle UWW \cong \triangle SWW \) (if we constructed it), but more likely, since we have \( UW \cong SW \) (from earlier steps, maybe step 8: \( SV = UW \) and \( SW \cong UV \) from CPCTC), \( VW \cong VW \) (reflexive, step 9), so by SSS, \( \triangle UWV \cong \triangle SWV \), then CPCTC gives \( \angle UWV \cong \angle SWV \). But in the table, step 10's reason should be CPCTC or SSS congruence. Wait, the last step (step 10) is to conclude \( \angle UWW \cong \angle SWW \), so the reason is CPCTC (if we used triangle congruence) or SSS. But from the given table, step 9 is \( \overline{VW} \cong \overline{VW} \) (reflexive), step 8 is \( SV = UW \), step 7 is \( SV = TU + TW \), etc. Wait, maybe the correct reason for step 10 is CPCTC (Corresponding Parts of Congruent Triangles are Congruent) because we have \( \triangle UWV \cong \triangle SWV \) by SSS ( \( UW \cong SW \), \( VW \cong VW \), \( UV \cong SW \) from CPCTC earlier), so the reason is CPCTC.

Answer:

The reason for step 10 (proving \( \angle UWW \cong \angle SWW \)) is CPCTC (Corresponding Parts of Congruent Triangles are Congruent). So the answer is to use CPCTC as the reason, meaning the statement \( \angle UWW \cong \angle SWW \) is true by CPCTC because the triangles containing these angles are congruent.