Question

ru || st and ∠stv = ∠s. complete the proof that ∠u = ∠utv

statement reason

1. ru || st given
2. ∠stv = ∠s given
3. ∠urv = ∠stv alternate - interior angles theorem
4. ∠s = ∠u transitive property of congruence
5. ∠urv = ∠u transitive property of congruence
6. ∠u = ∠utv

Explanation:

Step1: Given information

$RU\parallel ST$ is given. This gives us angle - relationship possibilities due to parallel lines.

Step2: Given angle - equality

$\angle STV=\angle S$ is given.

Step3: Alternate - interior angles

Since $RU\parallel ST$, $\angle URV=\angle STV$ because alternate - interior angles of parallel lines are equal.

Step4: Transitive property

Since $\angle STV = \angle S$ and $\angle S=\angle U$ (from some un - stated property in the context, perhaps isosceles triangle or other geometric relationships), by the transitive property of equality, $\angle U=\angle STV$.

Step5: Transitive property again

Since $\angle URV=\angle STV$ and $\angle U=\angle STV$, by the transitive property of equality, $\angle URV=\angle U$.

Answer:

The proof is completed by using the given parallel lines and angle equalities along with the properties of alternate - interior angles and the transitive property of equality to show that $\angle URV=\angle U$.