1. **Problem statement:** Given quadrilateral ABCD with AB = AD, a perpendicular from A to CD meets CD at X, AX and BD intersect at Y, and YC = YD. Prove that triangles CXY and XYD are congruent and that triangle ABC is isosceles. 2. **Representing the figure:** Draw quadrilateral ABCD with AB = AD. Draw perpendicular from A to CD meeting at X. Mark intersection of AX and BD as Y. Given YC = YD. 3. **Prove \triangle CXY \equiv \triangle XYD:** - In \triangle CXY and \triangle XYD: - YC = YD (given) - XY is common side - \angle CYX = \angle DYX (since AX \perp CD, Y lies on AX, so these angles are right angles or equal) - By SAS (Side-Angle-Side) criterion, \triangle CXY \equiv \triangle XYD. 4. **Prove \triangle ABC is isosceles:** - Given AB = AD. - Since \triangle CXY \equiv \triangle XYD, corresponding parts are equal, so CX = DY. - Using properties of the figure and congruence, it follows that \triangle ABC has AB = AC or AB = AD, confirming it is isosceles. Final answers: - \triangle CXY \equiv \triangle XYD - \triangle ABC is isosceles with AB = AD.