Final Answer:
Triangle NMK is congruent to triangle LKM due to the equality of side NK to side LK and the congruence of angle NKM to angle LKM.
Step-by-step explanation:
In geometry, the congruence of triangles is established by proving that their corresponding sides and angles are equal. In the case of triangles NMK and LKM, we can prove their congruence using the Side-Angle-Side (SAS) congruence criterion.
Firstly, we can establish that side NK is congruent to side LK. This can be observed by the definition of an isosceles triangle, where two sides are equal. In this scenario, MK acts as the base, and both NM and LK are the legs of their respective triangles, making NK and LK equal in length.
Secondly, angle NKM is congruent to angle LKM. Again, this is due to the nature of isosceles triangles, where the angles opposite the equal sides are also equal. Therefore, angle NKM and angle LKM are congruent.
Lastly, side MK is common to both triangles, and it remains the same in both. Thus, we have established the SAS criterion, proving that triangle NMK is congruent to triangle LKM.
In conclusion, the congruence of triangles NMK and LKM is confirmed by demonstrating that the corresponding sides and angles satisfy the SAS criterion. This logical and mathematical approach provides a solid foundation for understanding and applying congruence in geometric proofs.