1. **Problem Statement:** We have a parallelogram ABCD with diagonal BD. Points P and Q lie on sides AB and CD respectively such that angles labeled P2 and Q2 are equal. We need to prove: 7.1: Triangles APB and CQD are congruent. 7.2: Quadrilateral APQC is a parallelogram. 2. **Given:** - ABCD is a parallelogram. - BD is a diagonal. - Points P on AB and Q on CD such that $\angle P2 = \angle Q2$. 3. **Recall important properties:** - Opposite sides of a parallelogram are equal and parallel: $AB = DC$, $AD = BC$. - Diagonals bisect each other. - Congruence criteria for triangles: SAS, ASA, SSS, RHS. 4. **Proof for 7.1: $\triangle APB \equiv \triangle CQD$** - Since ABCD is a parallelogram, $AB \parallel DC$ and $AB = DC$. - Points P and Q lie on AB and DC respectively. - Given $\angle P2 = \angle Q2$. - Also, $BD$ is common to both triangles $APB$ and $CQD$. Check the triangles: - Side $AB = DC$ (opposite sides of parallelogram). - $\angle APB = \angle CQD$ (given as $\angle P2 = \angle Q2$). - Side $BD$ is common. By SAS (Side-Angle-Side) congruence criterion: $$\triangle APB \equiv \triangle CQD$$ 5. **Proof for 7.2: APQC is a parallelogram** - From 7.1, $\triangle APB \equiv \triangle CQD$ implies corresponding sides are equal. - So, $AP = CQ$ and $PB = QD$. - Since $AB \parallel DC$, and points P and Q lie on these sides, lines $AP$ and $QC$ are parallel. - Also, $AP = CQ$ and $PB = QD$ imply opposite sides of quadrilateral $APQC$ are equal and parallel. Therefore, $APQC$ is a parallelogram by definition. **Final answers:** 7.1 $\triangle APB \equiv \triangle CQD$ 7.2 $APQC$ is a parallelogram.