Error: △ KHL ≅ △ NHM by the Angle-Side-Angle Postulate. Correct Statement: △ KHL ≅ △ NHM by the Angle-Angle-Side Theorem.
Practice makes perfect
Let's analyze each step in the given proof and verify whether it is true. The fist two statements tell us that KH and NH 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 ∠ KHL is congruent to ∠ MHN 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 △ KHL and △ MHN 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 KH and NH 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, △ KHL and △ MHN can be proven to be congruent by the Angle-Angle-Side (AAS) Theorem. Let's correct this statement!
Statement
△ KHL ≅ △ MHN by AAS Theorem.
Then, from the fact that corresponding parts of congruent triangles are congruent, we have that LH ≅ MH. 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 LM. This is also true, as a midpoint divides a segment into two congruent segments.
