Question

∠rut≅∠sru. complete the proof that tv || qs.
statement: 1. ∠rut≅∠sru, 2. ∠prq≅∠sru, 3. ∠rut≅∠prq, 4. tv || qs
reason: 1. given, 2. vertical angle theorem, 3. transitive property of congruence

Explanation:

Step1: Identify given information

We are given that $\angle RUT\cong\angle SRU$.

Step2: Use vertical - angle theorem

$\angle PRQ$ and $\angle SRU$ are vertical angles. By the Vertical Angle Theorem, $\angle PRQ\cong\angle SRU$.

Step3: Apply transitive property

Since $\angle RUT\cong\angle SRU$ and $\angle PRQ\cong\angle SRU$, by the Transitive Property of Congruence, $\angle RUT\cong\angle PRQ$.

Step4: Use corresponding - angles postulate

$\angle RUT$ and $\angle PRQ$ are corresponding angles. If corresponding angles are congruent, then the lines are parallel. So, $\overleftrightarrow{TV}\parallel\overleftrightarrow{QS}$.

Answer:

The proof is completed by using the given angle - congruence, vertical - angle theorem, transitive property of congruence, and corresponding - angles postulate to show that $\overleftrightarrow{TV}\parallel\overleftrightarrow{QS}$.