Provide the reasons for the following proof.

Question

Given: JK = KL, NK = KM
Prove: ANKJ = AMLK

Answer 1

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.