Question

uw || rt. complete the proof that ∠tsv ≅ ∠swu.
statement reason

1. uw || rt given
2. ∠smu ≅ ∠qsr
3. ∠qsr ≅ ∠tsv
4. ∠tsv ≅ ∠swu transitive property of congruence

Explanation:

Step1: Identify corresponding - angles

Since $\overline{UW}\parallel\overline{RT}$, $\angle SWU$ and $\angle SRQ$ are corresponding angles. When two parallel lines are cut by a transversal ($\overline{QX}$), corresponding - angles are congruent. So the reason for $\angle SWU\cong\angle SRQ$ is "Corresponding angles postulate".

Step2: Identify vertical - angles

$\angle SRQ$ and $\angle TSV$ are vertical angles. Vertical angles are always congruent. So the reason for $\angle SRQ=\angle TSV$ is "Vertical angles are congruent".

Step3: Use transitive property

We have $\angle SWU\cong\angle SRQ$ and $\angle SRQ\cong\angle TSV$. By the transitive property of congruence, we can conclude that $\angle TSV\cong\angle SWU$.

Answer:

1. Corresponding angles postulate
2. Vertical angles are congruent