1. Problem 11: In triangle ABC, AB = AC, and points X and Y lie on AB and AC respectively such that AX = AY. Prove that \(\triangle ABY \cong \triangle ACX\). 2. Formula and rules: To prove two triangles congruent, we can use the SAS (Side-Angle-Side) criterion which states that if two sides and the included angle of one triangle are equal to the corresponding parts of another triangle, then the triangles are congruent. 3. Proof steps: - Given: AB = AC (triangle ABC is isosceles), AX = AY. - We want to prove \(\triangle ABY \cong \triangle ACX\). - Consider \(\triangle ABY\) and \(\triangle ACX\). - Side 1: AB = AC (given). - Side 2: AY = AX (given). - Angle: \(\angle BAY = \angle CAX\) because they are vertically opposite angles or angles at point A between segments AB and AC. - By SAS criterion, \(\triangle ABY \cong \triangle ACX\). Final answer: \(\triangle ABY \cong \triangle ACX\).