Question

complete the proof that $\triangle ruv\cong\triangle tsv$.

Explanation:

Step 1: Analyze ∠RVU and ∠SVT

∠RVU and ∠SVT are vertical angles. By the Vertical Angles Theorem, vertical angles are congruent. So the reason for ∠RVU ≅ ∠SVT is "Vertical Angles are Congruent".

Step 2: Prove Triangle Congruence

We have:

• \( \overline{ST} \cong \overline{RU} \) (Statement 1)
• \( \angle R \cong \angle T \) (Statement 4)
• \( \angle RVU \cong \angle SVT \) (Vertical Angles Theorem)

Using the Angle - Angle - Side (AAS) Congruence Criterion (if two angles and a non - included side of one triangle are congruent to the corresponding two angles and non - included side of another triangle, then the triangles are congruent), we can conclude that \( \triangle RUV\cong\triangle TSV \). The reason for \( \triangle RUV\cong\triangle TSV \) is "AAS Congruence Criterion" (since we have two angles (\( \angle R\cong\angle T \), \( \angle RVU\cong\angle SVT \)) and a non - included side (\( \overline{ST}\cong\overline{RU} \)) congruent).

Answer:

For statement 5, the reason is "Vertical Angles are Congruent". For statement 6, the reason is "AAS Congruence Criterion".