Question

Complete the proofs using the most appropriate method. Some may require CPCTC.

Given: KN = ZM, KN || LM
Prove: AKLN AMNL

Answer 1

Using the given information that KN = ZM and KN is parallel to LM, we can prove the triangles AKL and MNL are congruent by Angle-Side-Angle (ASA) congruence. Therefore, we conclude that triangle AKL is congruent to triangle MNL.

Step-by-step explanation:

To prove triangles AKL and MNL congruent, we utilize the information provided. The fact that KN = ZM establishes the corresponding sides of the two triangles. Furthermore, the parallel lines KN || LM create alternate interior angles, establishing the Angle-Side-Angle (ASA) congruence condition. The angles at K and L are congruent due to alternate interior angles formed by parallel lines.

Now, according to the ASA congruence criterion, when two angles and the included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent. In this case, ∠AKL is congruent to ∠MNL, KN = ZM, and ∠KAL is congruent to ∠MNL due to parallel lines. Therefore, by ASA, we can confidently assert that triangle AKL is congruent to triangle MNL.

Completing the proof, we can then state that the corresponding parts of congruent triangles are congruent (CPCTC). This implies that corresponding angles, sides, and vertices of triangles AKL and MNL are equal. Therefore, AKLN is congruent to AMNL. This logical progression solidifies the validity of the given statement and completes the proof.