1. **Problem:** Determine the measure of angle DC in quadrilateral ABCD given AB = 4, AD = 4, \(\angle B = 57^\circ\), and \(\angle D = 53^\circ\). 2. **Given:** Quadrilateral ABCD with sides AB = AD = 4, \(\angle B = 57^\circ\), \(\angle D = 53^\circ\). We want to find \(m\angle DC\) or determine if it can be found. 3. **Step:** Since AB = AD, triangle ABD is isosceles with base BD. Angles at B and D are given, but we do not have enough information about side BD or angle C. 4. **Step:** Without length BC or CD or additional angle information, we cannot apply the Law of Cosines or Law of Sines to find \(m\angle DC\). 5. **Conclusion:** The measure of angle DC cannot be determined from the given information. **Final answer:** Cannot be determined.