Question

complete the proof that qs || tv.
statement

1. ∠sru ≅ ∠rut
2. ∠prq ≅ ∠sru
3. ∠rut ≅ ∠prq
4. qs || tv

reason
given
vertical angle theorem

converse of corresponding angles theorem

Explanation:

Step1: Identify given equal - angles

We are given that $\angle SRU\cong\angle RUT$ (Statement 1).

Step2: Use vertical - angle property

By the Vertical Angle Theorem, $\angle PRQ\cong\angle SRU$ (Statement 2) since they are vertical angles.

Step3: Transitive property of congruence

Since $\angle RUT\cong\angle PRQ$ (Statement 3) because $\angle SRU\cong\angle RUT$ and $\angle PRQ\cong\angle SRU$.

Step4: Apply converse of corresponding - angles theorem

If $\angle RUT\cong\angle PRQ$, then by the Converse of the Corresponding Angles Theorem, $\overleftrightarrow{QS}\parallel\overleftrightarrow{TV}$ (Statement 4).

Answer:

$\overleftrightarrow{QS}\parallel\overleftrightarrow{TV}$ is proved using the given angle - congruences and angle - related theorems.