Question

complete the paragraph proof. given (overline{pr} parallel overline{kf}) and (angle ylr cong angle kxd) prove: (overline{wd} parallel overline{yh}) diagram of lines (wd), (yh), (pr), (kf) with points (w, v, x, d) on (wd); (y, l, z, h) on (yh); (p, v, l, r) on (pr); (k, x, z, f) on (kf) an incomplete paragraph proof shown includes some statements and reasons to prove (overline{wd} parallel overline{yh}). complete each missing statement/reason:
it is given that (\boxed{quad}). therefore, (angle ylr cong angle xzh) because if two parallel lines are cut by a transversal, then (\boxed{quad}). it is also given that (angle ylr cong angle kxd). it can be shown that (angle xzh cong angle kxd) by the (\boxed{quad}). therefore, it can be concluded that (overline{wd} parallel overline{yh}) because if two lines are intersected by a transversal so that (\boxed{quad}), then the lines are parallel.
options (partial): (overline{wd} parallel overline{yh}), (overline{pr} parallel overline{kf}), (angle xzh cong angle kxd), alternate exterior angles are congruent, corresponding angles are congruent, transitive property of congruence, definition of congruence...

Explanation:

Step1: Identify Given Parallel Lines

We are given \(\overline{PR} \parallel \overline{KF}\). When two parallel lines are cut by a transversal, corresponding angles are congruent. The transversal here for \(\overline{PR}\) and \(\overline{KF}\) can be \(\overline{YH}\) (or \(\overline{WD}\)). So, for \(\overline{PR} \parallel \overline{KF}\), \(\angle YLR\) and \(\angle XZH\) are corresponding angles, so \(\angle YLR \cong \angle XZH\) because corresponding angles are congruent.

Step2: Use Given Congruent Angles

We are also given that \(\angle YLR \cong \angle KXD\). Now, from Step 1, we have \(\angle YLR \cong \angle XZH\) and \(\angle YLR \cong \angle KXD\). By the Transitive Property of Congruence, if \(\angle YLR \cong \angle XZH\) and \(\angle YLR \cong \angle KXD\), then \(\angle XZH \cong \angle KXD\).

Step3: Prove Lines are Parallel

\(\angle XZH\) and \(\angle KXD\) are corresponding angles for lines \(\overline{WD}\) and \(\overline{YH}\) with transversal \(\overline{KF}\) (or \(\overline{PR}\)). If corresponding angles are congruent, then the lines are parallel. So, since \(\angle XZH \cong \angle KXD\) (corresponding angles), we can conclude that \(\overline{WD} \parallel \overline{YH}\).

Filling in the blanks:

1. First blank (It is given that): \(\boldsymbol{\overline{PR} \parallel \overline{KF}}\)
2. Second blank (transversal, then): \(\boldsymbol{\text{corresponding angles are congruent}}\)
3. Third blank (\(\angle XZH \cong \angle KXD\) by the): \(\boldsymbol{\text{Transitive Property of Congruence}}\)
4. Fourth blank (are intersected by a transversal so that): \(\boldsymbol{\text{corresponding angles are congruent}}\) (leading to \(\overline{WD} \parallel \overline{YH}\))

Answer:

1. \(\overline{PR} \parallel \overline{KF}\)
2. corresponding angles are congruent
3. Transitive Property of Congruence
4. corresponding angles are congruent (and thus \(\overline{WD} \parallel \overline{YH}\))