Question

1. (overline{nl}) is a diagonal of parallelogram (klmn). given
2. (overline{kl}paralleloverline{nm}) and (overline{kn}paralleloverline{lm}) definition of a parallelogram
3. ? ?
4. (overline{ln}congoverline{nl}) reflexive property of congruence
5. (\triangle klncong\triangle mnl) asa congruence criteria
6. (overline{kl}congoverline{nm}) and (overline{kn}congoverline{lm}) corresponding parts of congruent triangles are congruent

select the missing statement and reason to complete the given proof.
a. (angle lnkcongangle nlm), and (angle klncongangle mnl) by the alternate interior angles theorem
b. (angle klncongangle mln), and (angle knlcongangle mnl) by the alternate interior angles theorem
c. (angle lnkcongangle nlm), and (angle klncongangle mnl) by the corresponding angles theorem
d. (angle klncongangle mln), and (angle knlcongangle mnl) by the corresponding angles theorem

Explanation:

Since \( \overline{KL}\parallel\overline{NM}\) and \( \overline{KN}\parallel\overline{LM}\), when a transversal (\(\overline{LN}\)) intersects two parallel lines (\(\overline{KL}\) and \(\overline{NM}\), \(\overline{KN}\) and \(\overline{LM}\)), the alternate - interior angles are congruent.
For parallel lines \( \overline{KL}\parallel\overline{NM}\) and transversal \( \overline{LN}\), \( \angle KLN\cong\angle MLN\).
For parallel lines \( \overline{KN}\parallel\overline{LM}\) and transversal \( \overline{LN}\), \( \angle KNL\cong\angle MNL\).
The alternate - interior angles theorem states that if two parallel lines are cut by a transversal, then the alternate - interior angles are congruent.

Answer:

B. \( \angle KLN\cong\angle MLN\), and \( \angle KNL\cong\angle MNL\) by the alternate interior angles theorem