Triangle LNK is isosceles and line LK is the base. use a paragraph proof to prove that triangle LNK is equal to triangle KNM
To prove that triangle LNK is equal to triangle KNM, we can use the property that if a triangle has two sides congruent to two sides of another triangle, and the included angles are congruent, then the two triangles are congruent.
Since triangle LNK is isosceles, we know that the lengths LK and LK are congruent. Now, let's consider triangle KNM. From the statement of the problem, we can infer that line LK is the base of triangle LNK. Therefore, line NM must be the base of triangle KNM.
To prove that triangle LNK is equal to triangle KNM, we need to show that all three criteria for congruence are met.
1. Side-Side-Side (SSS) Criterion: We know that LK is congruent to LK (given) and NM is congruent to NM (since they are the same line). Hence, we have two pairs of congruent sides.
2. Angle-Angle-Side (AAS) Criterion: Since triangle LNK is isosceles, we can deduce that the angles LKN and LNK are congruent. Similarly, in triangle KNM, angles NKM and KNM are congruent, since triangle LNK and triangle KNM share the same angle, N. Therefore, we have two pairs of congruent angles.
3. Included Angle Criterion: The included angles for LNK and KNM are congruent because they share the side LK. Since LK is the base for both triangles, the included angles LKN and KNM are congruent.
Since all three criteria for congruence are satisfied, we conclude that triangle LNK is equal to triangle KNM, as desired.