Provide the reasons for the following proof.
Given: JK = KL, NK = KM
Prove: ANKJ = AMLK
The reasons for this proof are as follows:
1. Given statement: JK = KL
2. Given statement: NK = KM
3. Angles are congruent if and only if their corresponding sides are congruent. Since JK = KL, we can conclude that angle J is congruent to angle L.
4. Similarly, using the given statement NK = KM, we can conclude that angle N is congruent to angle M.
5. Triangle ANJ is congruent to triangle ALK by side-angle-side (SAS) congruence. This is because angle J is congruent to angle L (from reason 3), side JK is congruent to side KL (from reason 1), and side AN is congruent to side AL (common side).
6. Thus, we can write ANJ ≅ ALK.
7. By definition, the congruent triangles have congruent corresponding parts. Therefore, angle A in triangle ANJ is congruent to angle A in triangle ALK.
8. Similarly, angle N in triangle ANJ is congruent to angle L in triangle ALK.
9. Therefore, we can write ANKJ ≅ AMLK by the order of congruent parts (angle-angle-side, or AAS).
Thus, we have successfully proven that ANKJ is congruent to AMLK.