Question

1. is $\triangle bce \cong \triangle hie$? explain how you know. note: point $e$ is not the midpoint of $\overline{ci}$.

Explanation:

The problem asks if $\triangle BCE \cong \triangle HIE$. From the diagram: $\angle C$ and $\angle I$ are right angles (90°), $BC \cong HI$ (marked with equal segments), and $\angle BEC \cong \angle HEI$ (vertical angles). These satisfy the ASA (Angle-Side-Angle) congruence criterion: two angles and the included side of one triangle equal those of the other. Note: $E$ not being the midpoint of $\overline{CI}$ is irrelevant here—ASA only requires the included side (the right sides $BC/HI$) and the two angles around it (right angle + vertical angle).

Answer:

Yes, $\triangle BCE \cong \triangle HIE$ by ASA congruence.