1. **State the problem:** Given a quadrilateral ABCD with four equal sides (AB = BC = CD = DA) and one right angle (\(\angle DAB = 90^\circ\)), determine if ABCD is a square. 2. **Construct diagonal BD to create triangles:** Join BD to form triangles ADB and CDB. 3. **Prove congruency of triangles ADB and CDB:** - Sides: \(AD = CD\) (given), \(AB = CB\) (given), and \(BD\) is common. - By the Side-Side-Side (SSS) congruence criterion, \(\triangle ADB \cong \triangle CDB\). 4. **Conclude equal angles from congruency:** - From congruency, \(\angle C = \angle A = 90^\circ\). 5. **Analyze angles in \(\triangle DAB\):** - Let \(\angle 1 = \angle DAB\) and \(\angle 2 = \angle DBA\), with \(AB = AD\). - Since \(AB = AD\), \(\angle 1 = \angle 2\). - The sum of angles in \(\triangle DAB\) is \(\angle 1 + 90^\circ + \angle 2 = 180^\circ\). - Therefore, \(\angle 1 + \angle 2 = 90^\circ\). - So, \(\angle 1 = \angle 2 = 45^\circ\). 6. **Similarly analyze \(\triangle CDB\):** - Let \(\angle 3 = \angle DCB\) and \(\angle 4 = \angle DBC\) with \(CD = CB\). - Thus, \(\angle 3 = \angle 4\). - The sum of angles in \(\triangle CDB\) is \(\angle 3 + 90^\circ + \angle 4 = 180^\circ\). - Therefore, \(\angle 3 + \angle 4 = 90^\circ\). - So, \(\angle 3 = \angle 4 = 45^\circ\). 7. **Calculate remaining angles of the quadrilateral:** - \(\angle ABC = \angle 1 + \angle 4 = 45^\circ + 45^\circ = 90^\circ\). - \(\angle ADC = \angle 2 + \angle 3 = 45^\circ + 45^\circ = 90^\circ\). 8. **Conclusion on all four angles:** - Each angle of quadrilateral ABCD is \(90^\circ\). 9. **Final statement:** - With four equal sides and each angle being \(90^\circ\), ABCD is a square. 10. **Verification by measurement or SSS criterion:** - Measurement confirms \(AB = BC = CD = DA\) and all angles are right angles. - Using SSS congruence on triangles supports the same conclusion. **Answer:** The quadrilateral ABCD with four equal sides and one right angle is indeed a square.