Question

complete the proof that \\( \triangle ptv \cong \triangle rus \\).

1 \\( \angle s \cong \angle v \\)
2 \\( \angle ptv \cong \angle rus \\)
3 \\( \overline { s t } \cong \overline { u v } \\)
4 \\( t v = u v + t u \\)
5 \\( s u = s t + t u \\)
6 \\( t v = s t + t u \\)
7 \\( s u = t v \\)
8 \\( \triangle p t v \cong \triangle r u s \\)

Explanation:

Step1: Identify given information (angle equality)

From the diagram, $\angle S\cong\angle V$ is given as they are marked with the same symbol.

Step2: Vertical - angle related (angle equality)

$\angle PTV\cong\angle RUS$ because they are vertical angles. Vertical angles are congruent.

Step3: Given segment equality

$\overline{ST}\cong\overline{UV}$ is given as they are marked with the same symbol.

Step4: Segment addition postulate

$TV = UV+TU$ by the segment addition postulate (if a point $U$ lies on $\overline{TV}$, then $TV=TU + UV$).

Step5: Segment addition postulate

$SU = ST+TU$ by the segment addition postulate (if a point $T$ lies on $\overline{SU}$, then $SU=ST + TU$).

Step6: Substitution

Since $ST = UV$ (from step 3), substituting $UV$ for $ST$ in $TV = UV+TU$ (step 4) gives $TV = ST+TU$.

Step7: Transitive property of equality

From $SU = ST+TU$ (step 5) and $TV = ST+TU$ (step 6), by the transitive property of equality ($a=b$ and $a = c$ implies $b = c$), $SU=TV$.

Step8: ASA (Angle - Side - Angle) congruence

In $\triangle PTV$ and $\triangle RUS$, we have $\angle S\cong\angle V$ (step 1), $SU = TV$ (step 7), and $\angle PTV\cong\angle RUS$ (step 2). By the ASA (Angle - Side - Angle) congruence criterion, $\triangle PTV\cong\triangle RUS$.

Answer:

1. Given; 2. Vertical angles are congruent; 3. Given; 4. Segment addition postulate; 5. Segment addition postulate; 6. Substitution property (since \(ST = UV\)); 7. Transitive property of equality; 8. ASA (Angle - Side - Angle) congruence criterion.