1. **State the problem:** Given that \(\overline{BE} \cong \overline{FD}\), \(\overline{AE} \cong \overline{FC}\), and both \(\angle AEB\) and \(\angle CFD\) are right angles, prove that quadrilateral \(ABCD\) is a parallelogram. 2. **Recall the given information:** - \(\overline{BE} \cong \overline{FD}\) (Given) - \(\overline{AE} \cong \overline{FC}\) (Given) - \(\angle AEB\) and \(\angle CFD\) are right angles (Given) 3. **Use angle congruence:** - Since all right angles are congruent, \(\angle AEB \cong \angle CFD\). 4. **Apply the SAS congruence postulate:** - Triangles \(\triangle AEB\) and \(\triangle CFD\) have two pairs of congruent sides and the included angle congruent, so \(\triangle AEB \cong \triangle CFD\) by SAS. 5. **Use CPCTC (Corresponding Parts of Congruent Triangles are Congruent):** - From the congruent triangles, \(\overline{AB} \cong \overline{CD}\). 6. **Missing statement and reason:** - **Statement:** \(\angle BAE \cong \angle DCF\) - **Reason:** Corresponding parts of congruent triangles are congruent (CPCTC) 7. **Use alternate interior angles theorem:** - Since \(\angle BAE \cong \angle DCF\) and these are alternate interior angles formed by transversal \(\overline{AC}\) cutting lines \(\overline{AB}\) and \(\overline{CD}\), it follows that \(\overline{AB} \parallel \overline{CD}\). 8. **Conclude the proof:** - Quadrilateral \(ABCD\) has one pair of opposite sides that are both congruent and parallel, so \(ABCD\) is a parallelogram. **Final answer:** The missing statement is \(\angle BAE \cong \angle DCF\) and the reason is CPCTC.