In a triangle, ABC, AC = AB, and the altitude AD bisect BC. Prove that ΔADC ≅ ΔADB.

Suppose ABC is an isosceles triangle with AB = AC; BD and CE are bisectors of ∠B and ∠C. Prove that BD = CE.

Suppose ABC is an equiangular triangle. Prove that it is equilateral.(You have seen earlier that an equilateral triangle is equiangular. Thus for triangles equiangularity is equivalent to equilaterality.)

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?