Question

$\angle qst$ and $\angle wvx$ are supplementary. complete the proof that $\overleftrightarrow{rt}\parallel\overleftrightarrow{uw}$.

statement	reason
2 $m\angle qst+m\angle wvx = 180^{circ}$	definition of supplementary angles
3 $m\angle qst+m\angle tsv = 180^{circ}$	angles forming a linear pair sum to $180^{circ}$
4 $m\angle qst+m\angle wvx=m\angle qst+m\angle tsv$	transitive property of equality
5 $m\angle wvx=m\angle tsv$	properties of addition, subtraction, multiplication, and division
6

Explanation:

Step1: Analyze the given information

We know that \(\angle QST\) and \(\angle WVX\) are supplementary (\(m\angle QST + m\angle WVX=180^{\circ}\)) and \(\angle QST\) and \(\angle TSV\) form a linear - pair (\(m\angle QST + m\angle TSV = 180^{\circ}\)).

Step2: Use the transitive property

Since \(m\angle QST + m\angle WVX=m\angle QST + m\angle TSV\) (by the transitive property of equality as both equal \(180^{\circ}\)), we can subtract \(m\angle QST\) from both sides. Using the subtraction property of equality:
If \(a + b=a + c\), then \(b = c\). Here \(a=m\angle QST\), \(b = m\angle WVX\), \(c=m\angle TSV\), so \(m\angle WVX=m\angle TSV\).

Step3: Apply the corresponding - angles converse

If two lines are cut by a transversal and the corresponding angles are equal, then the lines are parallel. Here, \(\overrightarrow{RT}\) and \(\overrightarrow{UW}\) are cut by transversal \(\overrightarrow{QX}\), and \(\angle WVX\) and \(\angle TSV\) are corresponding angles. Since \(m\angle WVX=m\angle TSV\), by the converse of the corresponding - angles postulate, \(\overrightarrow{RT}\parallel\overrightarrow{UW}\).

Answer:

\(\overrightarrow{RT}\parallel\overrightarrow{UW}\) by the converse of the corresponding - angles postulate.