In the figure below, ABC DEF, A = 45°, C = 85°. Find the measures of D and E.

Since $\triangle ABC \equiv \triangle DEF$, their corresponding angles are equal.
Therefore, $\angle A = \angle D$.
Given $\angle A = 45^\circ$.

$$\angle D = 45^\circ$$

The sum of angles in a triangle is $180^\circ$.

$$\angle A + \angle B + \angle C = 180^\circ$$ $$45^\circ + \angle B + 85^\circ = 180^\circ$$ $$130^\circ + \angle B = 180^\circ$$ $$\angle B = 180^\circ - 130^\circ$$ $$\angle B = 50^\circ$$

Since $\triangle ABC \equiv \triangle DEF$, their corresponding angles are equal.
Therefore, $\angle B = \angle E$.
From Step 2, $\angle B = 50^\circ$.

$$\angle E = 50^\circ$$

The measures are: $\boxed{\angle D = 45^\circ, \angle E = 50^\circ}$