$\overline{st} \cong \overline{uv}$, $\angle v \cong \angle s$, and $\angle ptv \cong \angle rus$. complete the proof that $\delta ptv \cong \delta rus$.

$\begin{array}{l|l}
\text{statement} & \text{reason} \\
\hline
1.\ \overline{st} \cong \overline{uv} & \text{given} \\
2.\ \angle v \cong \angle s & \text{given} \\
3.\ \angle ptv \cong \angle rus & \text{given} \\
4.\ tv = uv + tu & \text{additive property of length} \\
5.\ su = st + tu & \text{additive property of length} \\
6.\ tv = st + tu & \text{substitution} \\
7.\ su = tv & \text{transitive property of equality} \\
8.\ &
\end{array}$

Question

$\overline{st} \cong \overline{uv}$, $\angle v \cong \angle s$, and $\angle ptv \cong \angle rus$. complete the proof that $\delta ptv \cong \delta rus$.

$\begin{array}{l|l}
\text{statement} & \text{reason} \\
\hline
1.\ \overline{st} \cong \overline{uv} & \text{given} \\
2.\ \angle v \cong \angle s & \text{given} \\
3.\ \angle ptv \cong \angle rus & \text{given} \\
4.\ tv = uv + tu & \text{additive property of length} \\
5.\ su = st + tu & \text{additive property of length} \\
6.\ tv = st + tu & \text{substitution} \\
7.\ su = tv & \text{transitive property of equality} \\
8.\ &
\end{array}$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Geometric Proofs", "Triangle Congruence" ], "new_concepts": [ "ASA Congruence" ], "current_concepts": [ "Geometric Proofs", "Triangle Congruence", "ASA Congruence" ] } ## Analyze the given proof steps Using the *Geometric Proofs* knowledge point $$ \begin{aligned} &\text{Given: } \overline{ST} \cong \overline{UV}, \quad \angle V \cong \angle S, \quad \angle PTV \cong \angle RUS\ &\text{Step 7: } SU = TV \implies \overline{SU} \cong \overline{TV} \end{aligned} $$ ## Determine the final congruence statement Using the *Triangle Congruence* knowledge point $$ \begin{aligned} &\text{In } \triangle PTV \text{ and } \triangle RUS:\ &\angle V \cong \angle S \quad (\text{Angle})\ &\overline{TV} \cong \overline{SU} \quad (\text{Side})\ &\angle PTV \cong \angle RUS \quad (\text{Angle})\ &\therefore \triangle PTV \cong \triangle RUS \quad (\text{by Angle-Side-Angle, ASA}) \end{aligned} $$ Complete the proof that $\triangle PTV \cong \triangle RUS$. 8. Statement: $\triangle PTV \cong \triangle RUS$ Reason: ASA { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Geometry", "Triangle Congruence" ] }

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QUESTION IMAGE

Question

Explanation:

Analyze the given proof steps

Determine the final congruence statement

Answer: