Question

∠wvx and ∠qst are supplementary. complete the proof that $overleftrightarrow{uw}paralleloverleftrightarrow{rt}$.
statement reason
1 ∠wvx and ∠qst are supplementary given
2 $m∠wvx + m∠qst=180^{circ}$ definition of supplementary angles
3 $m∠qst + m∠tsv = 180^{circ}$ angles forming a linear - pair sum to $180^{circ}$
4 $m∠wvx + m∠qst = m∠qst + m∠tsv$ transitive property of equality
5 $m∠wvx = m∠tsv$ properties of addition, subtraction, multiplication, and division
6 $overleftrightarrow{uw}paralleloverleftrightarrow{rt}$

Explanation:

Step1: State given information

Given $\angle WVX$ and $\angle QST$ are supplementary.

Step2: Use definition of supplementary angles

By definition, $m\angle WVX + m\angle QST=180^{\circ}$.

Step3: Identify linear - pair angles

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

Step4: Apply transitive property

Since $m\angle WVX + m\angle QST = 180^{\circ}$ and $m\angle QST + m\angle TSV = 180^{\circ}$, by the transitive property of equality, $m\angle WVX + m\angle QST=m\angle QST + m\angle TSV$.

Step5: Simplify the equation

Using properties of addition, subtraction, multiplication, and division (subtracting $m\angle QST$ from both sides), we get $m\angle WVX=m\angle TSV$.

Step6: Use corresponding - angles postulate

$\angle WVX$ and $\angle TSV$ are corresponding angles. If corresponding angles are congruent, then the lines are parallel. So, $\overleftrightarrow{UW}\parallel\overleftrightarrow{RT}$ (corresponding - angles postulate).

Answer:

Corresponding - angles postulate