Since the triangles $\triangle ABC$ and $\triangle DEF$ are congruent ($\triangle ABC \cong \triangle DEF$), all corresponding angles are equal. Given that $\angle C = 33^\circ$ in $\triangle ABC$, the measure of $\angle F$ in $\triangle DEF$ is also $33^\circ$.

Since $\triangle DEF$ is a right triangle (with $\angle D = 90^\circ$), we can determine $\angle E$ by using the angle sum property of a triangle:

$\angle D + \angle E + \angle F = 180^\circ$ $90^\circ + \angle E + 33^\circ = 180^\circ$ $\angle E = 180^\circ - 123^\circ = 57^\circ$

So, the measure of $\angle E$ is $57^\circ$.

Would you like further clarification on this solution?

1. How can congruent triangles help in solving for unknown angles?
2. What other methods could be used if the triangles were not congruent?
3. Can we determine side lengths using angle measures alone?
4. How would the solution change if the right angle was unknown?
5. Why is the angle sum property of a triangle essential?

Tip: Remember that congruent triangles have all corresponding angles and sides equal.