In a square PQRS, diagonals bisect each other at O. Prove that ΔPOQ ≅ ΔQOR ≅ ΔROS ≅ ΔSOP.

In the figure, two sides AB, BC and the median AD of ΔABC are respectively equal to two sides PQ, QR and median PS of ΔPQR. Prove that

(i) ΔADB ≅ ΔPSQ;

(ii) ΔADC ≅ ΔPSR.

Does it follow that triangles ABC and PQR are congruent?

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.

In ΔPQR, PQ = QR; L, M, and N are the midpoints of the sides of PQ, QR and RP respectively. Prove that LN = MN.

In a square PQRS, diagonals bisect each other at O. Prove that ΔPOQ ≅ ΔQOR ≅ ΔROS ≅ ΔSOP.

In the figure ABCD is a square, M, N, O, and P are the midpoints of sides AB, BC, CD and DA respectively. Identify the congruent triangles.

In the adjoining figure, PQRS is a rectangle. Identify the congruent triangles formed by the diagonals.

In the adjoining figure, the sides BA and CA have been produced such that BA = AD and CA = AE. Prove that DE || BC. [ hint:-use the concept of alternate angles.]

Suppose a triangle is equilateral. Prove that it is equiangular.

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.