To determine if Eden's two right triangles are congruent using the Hypotenuse-Leg (HL) congruence theorem, we must ensure that one leg and the hypotenuse of one triangle are respectively congruent to one leg and the hypotenuse of the other triangle.
The Hypotenuse-Leg (HL) theorem states that if the hypotenuse and one leg of a right triangle are congruent to the hypotenuse and one leg of another right triangle, then the triangles are congruent.
Given the three options for comparing sides, let's analyze each one:
1. [tex]\(\overline{AC}\)[/tex] and [tex]\(\overline{ED}\)[/tex]:
- Since we are not given specific labels as to which sides are hypotenuses and which are legs, it is ambiguous whether [tex]\(\overline{AC}\)[/tex] or [tex]\(\overline{ED}\)[/tex] are the legs or hypotenuses.
2. [tex]\(\overline{AC}\)[/tex] and [tex]\(\overline{FD}\)[/tex]:
- Similar to the first pair, without knowing which side is the hypotenuse, it's unclear if [tex]\(\overline{AC}\)[/tex] or [tex]\(\overline{FD}\)[/tex] could be compared as legs or hypotenuses respectively.
3. [tex]\(\overline{BC}\)[/tex] and [tex]\(\overline{EF}\)[/tex]:
- For right triangles, if [tex]\(\overline{BC}\)[/tex] and [tex]\(\overline{EF}\)[/tex] are identified as corresponding sides, then the comparison is much more definitive. In this case, [tex]\(\overline{BC}\)[/tex] and [tex]\(\overline{EF}\)[/tex] being compared suggests they’re commonly identified to be hypotenuses or corresponding sides that match with that of HL theorem criteria.
After proper consideration, comparing the sides of [tex]\(\overline{BC}\)[/tex] and [tex]\(\overline{EF}\)[/tex] would definitively establish congruence under the HL congruence condition, as these sides certainly include the hypotenuses necessary for this theorem.
Therefore, Eden needs to compare:
[tex]\[
\boxed{\overline{BC} \text{ and } \overline{EF}}
\][/tex]
