In parallelogram KLMN, opposite sides are congruent (KL = MN) and alternate interior angles are equal (KNL = MLN). Since KMN is isosceles (KL = MN), base angles KNM and LMN are congruent. So, KNL ≅ MLN by Angle-Angle-Angle (AA) congruence.
Opposite sides of a parallelogram are congruent: This is a property of parallelograms. In this case, sides KL and MN are opposite sides, so KL = MN.
Alternate interior angles of parallel lines are congruent: Since lines KN and LM are parallel (opposite sides of a parallelogram are parallel), and lines KL and MN are cut by transversal KM, angles KNL and MLN are alternate interior angles. Therefore, KNL = MLN.
Base angles of an isosceles triangle are congruent Since KL = MN (from step 1), triangle KMN is an isosceles triangle. Therefore, angles KNM and LMN are congruent.
Now we have that:
KNL = MLN (from step 2)
LMN = KNM (from step 3)
By the Angle-Angle-Angle (AA) congruence criterion, triangle KNL is congruent to triangle MLN.
Therefore, we have proven that triangle KNL is congruent to triangle MLN.