Question

Given that KJ = MN and that L is the midpoint of JN, prove JKL = NML.

Answer 1

Let's follow the statements in the figure you provide and give the corresponding reason for each.

1. KJ is congruent to MN ---> GIVEN (That means that you were told this is true when the problem was posed. Read the problem...everything before the word "prove" is given to you).

2. JL is congruent to LN --> You were told that L is the midpoint of JN. A midpoint divides a segment into two congruent parts. So, the fact that L is a midpoint means that JL and LN are congruent. Here we do not say given because you were given that L is the midpoint and you had to now what a midpoint does to get to the fact that JL and LN are congruent. So here we say DEFINITION OF A MIDPOINT

3. angle JKL is congruent to angle NLM --> GIVEN. Here the information was given to you in the diagram. These angles are both drawn with a red curved line. Since these are the same (both a single red curve) it means that the angles are congruent.

PROVING TRIANGLES CONGRUENT
We are asked to prove that triangle JKL and triangle NML are congruent. There are various ways to prove that two triangles are congruent. These are:
SSS (each side in one triangle is congruent to a side in the other triangle)
SAS (two sides and the angle between them in one triangle are congruent to two sides and the angle between them in the other triangle)
ASA (two angles and the side between them in one triangle are congruent to two angles and the side between them in the other triangle)
AAS Two angles and a side not between them in one triangle is congruent to two angles and a side not between them in the other triangle.
HL (the hypotenuse and a leg of a right triangle is congruent to the hypotenuse and a leg in another right triangle)

OUR TRIANGLES
Attached is a diagram for this problem. In it since KJ and MN are congruent I have marked each with a single hashmark. Since JL and LN are congruent I have marked each of these with two hashmarks. You will notice that both triangles have a side with one hashmark and another with two hashmarks Also, there is an angle congruent in each. However, the angle is not BETWEEN the sides so we cannot use SAS. None of the other methods of proving two triangles congruent work because what we have is two sides and an angles not between them. That would be SSA but SSA is NOT one of the ways of proving triangles congruent. The triangles we are given might be congruent but they also might not be.

The answers is D --> There is not enough information to prove the triangles congruent.