**(a) In figure (1) given below, ABCD is a parallelogram and X is mid-point of BC. The line AX produced meets DC produced at Q. The parallelogram ABPQ is completed. Prove that:**

**(i) the triangles ABX and QCX are congruent;**

**(ii)DC = CQ = QP**

**(b) In figure (2) given below, points P and Q have been taken on opposite sides AB and CD respectively of a parallelogram ABCD such that AP = CQ. Show that AC and PQ bisect each other.**

**Solution:**

**More Solutions:**

- Prove that 3 ∠POB = ∠AOP.
- Prove that EFGH is a square.
- ABCD and ABEF are parallelograms.
- Find the ratio AC : BD.
- ABCD in which ∠ BAD = 45°
- AB = 6cm, BC = 4cm, CD = 4 cm
- Construct the quadrilateral ABCD.
- Which AB = 3.3 cm, BC = 4.9 cm, CD = 5.8 cm.
- Construct a trapezium ABCD in which AD || BC
- AB = 5 cm. BC = 6.2 cm and CD = 4.8 cm.