Answer
(a) $\triangle ABE$ ~ $\triangle CFE$
(b) $\triangle CFE$ ~ $\triangle DFC$
(c) $\triangle ABE$ ~ $\triangle DFC$
Work Step by Step
(a) We can find $m\angle CFE$:
$m\angle CFE + m\angle CFD = 180^{\circ}$
$m\angle CFE + 90^{\circ} = 180^{\circ}$
$m\angle CFE = 90^{\circ}$
We can find two congruent angles in $\triangle ABE$ and $\triangle CFE$:
$\angle ABE \cong \angle CFE$, since both angles are $90^{\circ}$
$\angle BEA \cong \angle FEC$, since these angles are opposite angles
Since there are two congruent angles in $\triangle ABE$ and $\triangle CFE$, all three angles must be congruent.
Therefore, $\triangle ABE$ ~ $\triangle CFE$
(b) We can find two congruent angles in $\triangle CFE$ and $\triangle DFC$:
$\angle CFE \cong \angle DFC$, since both angles are $90^{\circ}$
$\angle FCE \cong \angle CDF$, since $m\angle FCE = 90^{\circ} - m\angle FCD = m\angle CDF$
Since there are two congruent angles in $\triangle CFE$ and $\triangle DFC$, all three angles must be congruent.
Therefore, $\triangle CFE$ ~ $\triangle DFC$
(c) $\triangle ABE$ ~ $\triangle CFE$ ~ $\triangle DFC$
Therefore, $\triangle ABE$ ~ $\triangle DFC$
