Let's analyze each step in the given proof and verify whether it is true. The fist two statements tell us that $\overline{K H}$ and $\overline{N H}$ are congruent sides, and that $∠ L$ and $∠ M$ are congruent angles. Since this information is given, we know that these two statements are correct.

The next statement says that $∠ K H L$ is congruent to $∠ M H N$ because they are vertical angles. Analyzing the diagram, we can see that these angles are indeed vertical angles. Moreover, the Vertical Angles Theorem ensures us that they are congruent. We can conclude that this statement is also correct.

Next, it is given that $△ K H L$ and $△ M H N$ are congruent by Angle-Side-Angle (ASA) Postulate. Let's recall what this postulate states.

Angle-Side-Angle (ASA) Postulate
If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then two triangles are congruent.

In our case, the congruent sides $\overline{K H}$ and $\overline{N H}$ are not included between the congruent angles. Therefore, the ASA Postulate cannot be used to prove the triangles congruent. Let's recall the Angle-Angle-Side (AAS) Theorem, which relates two pairs of angles and a non-included side.

Angle-Angle-Side (AAS) Theorem
If two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of another triangle, then two triangles are congruent.

Therefore,

△ K H L

and

△ M H N

can be proven to be congruent by the Angle-Angle-Side (AAS) Theorem. Let's correct this statement!

\underline{Statement} △ K H L ≅ △ M H N by AAS Theorem .

Then, from the fact that corresponding parts of congruent triangles are congruent, we have that

\overline{L H} ≅ \overline{M H} .

This statement is true, as these sides are indeed corresponding. The last statement says that, by the definition of a midpoint,

H

is the midpoint of

\overline{L M} .

This is also true, as a midpoint divides a segment into two congruent segments.

Exercise