1. **State the problem:** We are given two line segments $\overline{DE}$ and $\overline{FG}$ intersecting at point $H$, where $H$ is the midpoint of both segments. We know $\angle 2$ is complementary to $\angle 3$, and $\angle 3$ is congruent to $\angle 4$. We need to prove that $\angle 2 \cong \angle 4$. 2. **Recall definitions and properties:** - Since $H$ is the midpoint of both $\overline{DE}$ and $\overline{FG}$, the segments $\overline{DH} \cong \overline{HE}$ and $\overline{FH} \cong \overline{HG}$. - Complementary angles sum to 90 degrees: $\angle 2 + \angle 3 = 90^\circ$. - Given $\angle 3 \cong \angle 4$, so they have equal measures. 3. **Use the given information:** Since $\angle 2 + \angle 3 = 90^\circ$ and $\angle 3 = \angle 4$, substitute $\angle 4$ for $\angle 3$: $$\angle 2 + \angle 4 = 90^\circ$$ 4. **Analyze the angles:** Because $\angle 3$ and $\angle 4$ are congruent and $\angle 2$ is complementary to $\angle 3$, $\angle 2$ must also be complementary to $\angle 4$. 5. **Prove congruence:** If $\angle 2 + \angle 4 = 90^\circ$ and $\angle 3 = \angle 4$, then $\angle 2$ and $\angle 4$ are complementary to the same angle $\angle 3$, implying $\angle 2 \cong \angle 4$ by the property of congruent complements. **Final answer:** $$\boxed{\angle 2 \cong \angle 4}$$