Question

proof: 1. \\( \overline { \mathrm { nl } } \\) is a diagonal of parallelogram klmn given 2. \\( \overline { \mathrm { kl } } \parallel \overline { \mathrm { nm } } \\) and \\( \overline { \mathrm { kn } } \parallel \overline { \mathrm { lm } } \\) definition of a parallelogram 3. ? ? 4. \\( \overline { \mathrm { ln } } \cong \overline { \mathrm { nl } } \\) reflexive property of congruence 5. \\( \delta \mathrm { kln } \cong \delta \mathrm { mnl } \\) asa congruence criteria 6. \\( \overline { \mathrm { kl } } \cong \overline { \mathrm { nm } } \\) and \\( \overline { \mathrm { kn } } \cong \overline { \mathrm { lm } } \\) corresponding parts of congruent triangles are congruent select the missing statement and reason to complete the given proof. a. \\( \angle \mathrm { lnk } \cong \angle \mathrm { nlm } \\), and \\( \angle \mathrm { kln } \cong \angle \mathrm { mnl } \\) by the corresponding angles theorem b. \\( \angle \mathrm { kln } \cong \angle \mathrm { mln } \\), and \\( \angle \mathrm { knl } \cong \angle \mathrm { mnl } \\) by the corresponding angles theorem c. \\( \angle \mathrm { kln } \cong \angle \mathrm { mln } \\), and \\( \angle \mathrm { knl } \cong \angle \mathrm { mnl } \\) by the alternate interior angles theorem d. \\( \angle \mathrm { lnk } \cong \angle \mathrm { nlm } \\), and \\( \angle \mathrm { kln } \cong \angle \mathrm { mnl } \\) by the alternate interior angles theorem

Explanation:

Step1: Recall properties of parallel lines

When two parallel lines are cut by a transversal, alternate - interior angles are congruent. In parallelogram \(KLMN\), \(\overline{KL}\parallel\overline{NM}\) and \(\overline{KN}\parallel\overline{LM}\), and \(\overline{NL}\) is a transversal.

Step2: Identify alternate - interior angles

For \(\overline{KL}\parallel\overline{NM}\) and transversal \(\overline{NL}\), \(\angle KLN\cong\angle MLN\). For \(\overline{KN}\parallel\overline{LM}\) and transversal \(\overline{NL}\), \(\angle KNL\cong\angle MNL\). These angle congruences are by the alternate - interior angles theorem.

Answer:

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