Question

complete the proof that \\(\overleftrightarrow{tv} \parallel \overleftrightarrow{qs}\\).
(image of a diagram with lines and angles, and a table with statements and reasons:

1. \\(\angle rut \cong \angle sru\\) (given)
2. \\(\angle prq \cong \angle sru\\) (reason blank)
3. \\(\angle rut \cong \angle prq\\) (reason blank)
4. \\(\overleftrightarrow{tv} \parallel \overleftrightarrow{qs}\\) (reason blank))

Explanation:

Step1: Identify vertical angles

$\angle PRQ\cong\angle SRU$ because vertical angles are congruent.

Step2: Use transitive property

Since $\angle RUT\cong\angle SRU$ (given) and $\angle PRQ\cong\angle SRU$, by the transitive property of congruence, $\angle RUT\cong\angle PRQ$.

Step3: Apply converse of corresponding angles postulate

If two lines are cut by a transversal and the corresponding angles are congruent, then the lines are parallel. Here, $\overleftrightarrow{TV}$ and $\overleftrightarrow{QS}$ are cut by transversal $\overleftrightarrow{WP}$, and $\angle RUT$ (corresponding to $\angle PRQ$) are congruent. So, by the converse of the corresponding angles postulate, $\overleftrightarrow{TV}\parallel\overleftrightarrow{QS}$.

Answer:

1. Vertical angles are congruent; 3. Transitive property of congruence; 4. Converse of corresponding angles postulate.