Let ABC be a triangle. Draw a triangle BDC externally on BC such that AB = BD and AC = CD. Prove that ΔABC ≅ ΔDBC.

Two triangles ABC and DBC have common base BC. Suppose AB = DC and ∠ABC = ∠BCD. Prove that AC = BD.

Let AB and CD be two line segments such that AD and BC intersect at O. Suppose AO = OC and BO = OD. Prove that AB = CD.

Let ABCD be a square and let points P on AB and Q on DC be such that DP = AQ. Prove that BP = CQ.

In a triangle ABC, AB = AC. Suppose P is a point on AB and Q is a point on AC such that AP = AQ. = Prove that ΔAPC ≅ ΔAQB.