Question

complete the paragraph proof. given: (overline{wd} parallel overline{yh}) and (angle pvx cong angle xzh) prove (overline{pr} parallel overline{kf}) an incomplete paragraph proof shown includes some statements and reasons to prove (overline{pr} parallel overline{kf}). complete each... it is given that... therefore, (angle pvx cong angle vlz) because... transversal, then... it is also given that (angle pvx cong angle xz)... it can be written that (angle vlz cong angle xzh). therefore, if... if two lines are intersected by a transversal so that... then...

Explanation:

Step1: Identify Given Parallel Lines

We know \( \overline{WD} \parallel \overline{YH} \) (given). When two parallel lines are cut by a transversal, corresponding angles are congruent. So, for transversal \( PR \) (or the line containing \( PR \)) cutting \( WD \) and \( YH \), \( \angle PVX \cong \angle VLZ \) (corresponding angles postulate).

Step2: Use Given Congruent Angles

We are also given \( \angle PVX \cong \angle XZH \). By the transitive property of congruence, since \( \angle PVX \cong \angle VLZ \) and \( \angle PVX \cong \angle XZH \), we have \( \angle VLZ \cong \angle XZH \).

Step3: Prove \( PR \parallel KF \)

Now, \( \angle VLZ \) and \( \angle XZH \) are corresponding angles formed by transversal \( YH \) (or the line containing \( YH \)) cutting \( PR \) and \( KF \). If corresponding angles are congruent, then the lines are parallel. So, by the converse of the corresponding angles postulate, \( \overline{PR} \parallel \overline{KF} \).

Answer:

To complete the proof:

1. First, since \( \boldsymbol{\overline{WD} \parallel \overline{YH}} \) (given), \( \angle PVX \cong \angle VLZ \) because corresponding angles are congruent when parallel lines are cut by a transversal.
2. Given \( \angle PVX \cong \angle XZH \), by transitivity, \( \angle VLZ \cong \angle XZH \).
3. Finally, since \( \angle VLZ \) and \( \angle XZH \) are corresponding angles for lines \( PR \) and \( KF \) cut by transversal \( YH \), and they are congruent, by the converse of the corresponding angles postulate, \( \boldsymbol{\overline{PR} \parallel \overline{KF}} \).

(Note: The specific blanks in the problem’s proof would be filled with:

• First blank: \( \overline{WD} \parallel \overline{YH} \)
• Reason for \( \angle PVX \cong \angle VLZ \): corresponding angles are congruent
• Transitive step: \( \angle VLZ \cong \angle XZH \)
• Final conclusion: \( \overline{PR} \parallel \overline{KF} \) via converse of corresponding angles postulate)