1. **State the problem:** Prove that a parallelogram with congruent diagonals has a right angle. 2. **Given:** - \(\overline{AC} \cong \overline{DB}\) (diagonals are congruent) - \(\overline{AB} \parallel \overline{CD}\) and \(\overline{AD} \parallel \overline{BC}\) (definition of parallelogram) 3. **Recall theorem:** Opposite sides in a parallelogram are congruent. 4. **Step 1:** From the parallelogram properties, \(\overline{AB} \cong \overline{DC}\) and \(\overline{AD} \cong \overline{BC}\). 5. **Step 2:** \(\overline{AD} \cong \overline{AD}\) (segment congruent to itself). 6. **Step 3:** Given \(\overline{AC} \cong \overline{DB}\). 7. **Step 4:** By Side-Side-Side (SSS) congruence, triangles \(\triangle ABE\) and \(\triangle CDE\) are congruent (using sides \(AB \cong DC\), \(AD \cong AD\), and \(AC \cong DB\)). 8. **Step 5:** Corresponding angles \(\angle 1\) and \(\angle 2\) are congruent by CPCTC (Corresponding Parts of Congruent Triangles are Congruent). 9. **Step 6:** Since \(\overline{AB} \parallel \overline{CD}\) and \(\overline{AD} \parallel \overline{BC}\), angles \(\angle 1\) and \(\angle 4\) are same-side interior angles formed by transversal \(\overline{AD}\). 10. **Step 7:** Same-side interior angles sum to \(180^\circ\), so \(m\angle 1 + m\angle 4 = 180^\circ\). 11. **Step 8:** Because \(\angle 1 \cong \angle 4\) (from congruent triangles), substitute to get \(2 \cdot m\angle 1 = 180^\circ\). 12. **Step 9:** Divide both sides by 2 to find \(m\angle 1 = 90^\circ\). **Conclusion:** The parallelogram has a right angle, proving the statement.