1. **Problem 11:** In triangle ABC, AB = AC, points X and Y lie on AB and AC respectively such that AX = AY. Prove that \(\triangle ABY \cong \triangle ACX\). 2. **Formula and rule:** To prove two triangles congruent, we can use the SAS (Side-Angle-Side) criterion: if two sides and the included angle of one triangle are equal to the corresponding parts of another triangle, the triangles are congruent. 3. **Step-by-step solution for 11:** - Given: AB = AC, AX = AY. - Since X lies on AB and Y lies on AC, segments BX and CY are parts of AB and AC respectively. - Angles \(\angle BAX = \angle CAY\) because AX = AY and they are corresponding segments from the vertex A. - In \(\triangle ABY\) and \(\triangle ACX\): - Side AB = AC (given) - Side AY = AX (given) - Angle \(\angle BAX = \angle CAY\) (common angle) - By SAS criterion, \(\triangle ABY \cong \triangle ACX\). 4. **Problem 12:** In isosceles triangle LMN with LM = LN, LO is the angle bisector of \(\angle MLN\). Prove O is the midpoint of MN. 5. **Step-by-step solution for 12:** - Given: LM = LN, LO bisects \(\angle MLN\). - In \(\triangle LMO\) and \(\triangle LNO\): - LM = LN (given) - LO = LO (common side) - \(\angle MLO = \angle NLO\) (LO bisects \(\angle MLN\)) - By SAS, \(\triangle LMO \cong \triangle LNO\). - Therefore, corresponding parts are equal: MO = NO. - Hence, O is the midpoint of MN. 6. **Problem 13:** In triangle PQR, LM = MN, QM = MR, ML \perp PQ, MN \perp PR. Prove PQ = PR. 7. **Step-by-step solution for 13:** - Given: LM = MN, QM = MR, ML \perp PQ, MN \perp PR. - Consider right triangles formed by these perpendiculars. - Since LM = MN and QM = MR, triangles formed are congruent by RHS (Right angle-Hypotenuse-Side) criterion. - Therefore, corresponding sides PQ and PR are equal. 8. **Problem 10 (Assertion and Reason):** - Assertion (A): In quadrilateral PQRS, PR = PS and PQ bisects \(\angle P\) by SAS congruency. - Reason (R): SAS congruency axiom states if two sides and the included angle of one triangle equal the corresponding parts of another, the triangles are congruent. - This is true because the SAS criterion is a fundamental rule for triangle congruency. **Final answers:** - 11: \(\triangle ABY \cong \triangle ACX\) by SAS. - 12: O is midpoint of MN. - 13: PQ = PR. - 10: Assertion and Reason are correct by SAS congruency axiom.