Question

statement\treason
1 ∠qsr and ∠uvx are supplementary\tgiven
2 m∠qsr + m∠uvx = 180°\tdefinition of supplementary angles
3 m∠qsr + m∠rsv = 180°\tangles forming a linear pair sum to 180°
4 m∠qsr + m∠uvx = m∠qsr + m∠rsv\tproperties of addition, subtraction, multiplication, and division
5 m∠uvx = m∠rsv
6 uw || rt\tconverse of corresponding angles theorem

Explanation:

Step1: Start with given information

Given that $\angle QSR$ and $\angle UVX$ are supplementary, so $m\angle QSR + m\angle UVX=180^{\circ}$ by the definition of supplementary angles.

Step2: Use linear - pair property

Since $\angle QSR$ and $\angle RSV$ form a linear pair, $m\angle QSR + m\angle RSV = 180^{\circ}$ (angles forming a linear pair sum to $180^{\circ}$).

Step3: Set up equalities

We have $m\angle QSR + m\angle UVX=m\angle QSR + m\angle RSV$ (by properties of addition, subtraction, multiplication, and division, specifically subtracting $m\angle QSR$ from both sides of the equalities).

Step4: Simplify to get angle - equality

Subtracting $m\angle QSR$ from both sides of $m\angle QSR + m\angle UVX=m\angle QSR + m\angle RSV$ gives $m\angle UVX = m\angle RSV$.

Step5: Apply converse of corresponding - angles theorem

Since $m\angle UVX = m\angle RSV$, by the converse of the corresponding - angles theorem, $\overleftrightarrow{UW}\parallel\overleftrightarrow{RT}$.

Answer:

The proof shows that $\overleftrightarrow{UW}\parallel\overleftrightarrow{RT}$ based on the given information about supplementary angles, linear - pair angles, and the converse of the corresponding - angles theorem.