[Math] Trying to prove that two angles are congruent in a isosceles trapezoid

Question

I was given this assignment to do the following.

Write a paragraph proof for the following scenario.

Given: $ KLMN $ is an isosceles trapezoid.

Prove: $ \angle LKM \cong \angle MNL $

The thing is that I think I successfully did this, but the actual answer is way different from mine. It’s more complicated, actually. I’d like someone to please review this and tell me if:

1. Is this proof correct?
2. If not, then why not?

The handwritten text above is:

It is given that $ KLN $ is an isosceles trapezoid. Both $ \overline{ LK } $ and $ \overline{ MN } $ are congruent due to the definition of an isosceles trapezoid. $ \angle L $ and $ \angle M $ are congruent; angles on the either sides of the bases of an isosceles trapezoid are congruent is a property.¹ Vertical angle theorem states that vertical angles are congruent, so angles X and Y are congruent. $ \triangle LXH \cong \triangle MYN $ due to AAS.² $ \angle K \cong \angle N $, because corresponding parts of congruent triangles are congruent.

¹_{Originally transcribed by OP as is another property as well}

²_{Amplified in original transcription by OP to Angle-Angle-Side Theorem}

Answer 1

You start correctly with "both $\overline{LK}$ and $\overline{MN}$ are congruent". (As a style point, in UK usage you would phrase this "The lines $\overline{LK}$ and $\overline{MN}$ are equal in length." It's a bit clearer to keep "congruent" for referring to triangles or other two-or-more-dimensional shapes.)

The other property you are given is that $\angle L$ and $\angle M$ are equal. This means that the angles between the edges of the trapezoid are at those corners are equal, $\widehat{KLM}=\widehat{LMN}$. But none of your working mentions these angles clearly and you don't mark them on any of your diagrams. So this is where your answer starts going wrong.

(Finally, and after the error: the sentence "angles on the either sides of the bases of an isosceles trapezoid are congruent is another property as well" doesn't make much sense to me. Do you mean "$\angle N = \angle K$"? That's easily proved using "alternate angles".)