Question

3.8.0 check your understanding
complete the paragraph proof.
given: $overline{wd}paralleloverline{yh}$ and $angle pvxcongangle xzh$
prove: $overline{pr}paralleloverline{kf}$
an incomplete paragraph proof shown includes some statements and reasons to prove $overline{pr}paralleloverline{kf}$. complete each missing statement.
it is given that
therefore, $angle pvxcongangle vlz$ because if two parallel lines are intersected by a transversal, then

Explanation:

Step1: Recall parallel - line properties

Since \(WD\parallel YH\), by the corresponding - angles postulate, when two parallel lines are intersected by a transversal, corresponding angles are congruent. So, \(\angle PVX\cong\angle VLZ\) because \(\angle PVX\) and \(\angle VLZ\) are corresponding angles with respect to the parallel lines \(WD\) and \(YH\) and the transversal \(PR\).

Step2: Use the given congruence

We are given that \(\angle PVX\cong\angle XZH\). By the transitive property of congruence, if \(\angle PVX\cong\angle VLZ\) and \(\angle PVX\cong\angle XZH\), then \(\angle VLZ\cong\angle XZH\).

Step3: Apply the converse of corresponding - angles postulate

The converse of the corresponding - angles postulate states that if two lines are cut by a transversal and the corresponding angles are congruent, then the two lines are parallel. Since \(\angle VLZ\) and \(\angle XZH\) are corresponding angles with respect to the lines \(PR\) and \(KF\) and the transversal \(YH\), and \(\angle VLZ\cong\angle XZH\), we can conclude that \(PR\parallel KF\).

Answer:

The proof is completed by using the corresponding - angles postulate for parallel lines \(WD\) and \(YH\), the transitive property of congruence, and the converse of the corresponding - angles postulate to show that \(PR\parallel KF\).