Some More Criteria for Congruence of Triangles

ΔABC and ΔDBC are two isosceles triangles on the same base BC and vertices A and D are on the same side of BC (see the given figure). If AD is extended to intersect BC at P, show that

1. ΔABD ≅ ΔACD
2. ΔABP ≅ ΔACP
3. AP bisects ∠A as well as ∠D.
4. AP is the perpendicular bisector of BC.

ABC and DBC are two triangles on the same base BC such that A and D lie on the opposite sides of BC, AB = AC and DB = DC. Show that AD is the perpendicular bisector of BC.

In the following figure, BA ⊥ AC, DE ⊥ DF such that BA = DE and BF = EC. Show that ∆ABC ≅ ∆DEF.

AD is an altitude of an isosceles triangles ABC in which AB = AC. Show that

1. AD bisects BC
2. AD bisects ∠A

ABCD is quadrilateral such that AB = AD and CB = CD. Prove that AC is the perpendicular bisector of BD.

Prove that sum of any two sides of a triangle is greater than twice the median with respect to the third side.

Two lines l and m intersect at the point O and P is a point on a line n passing through the point O such that P is equidistant from l and m. Prove that n is the bisector of the angle formed by l and m.

ABC is a right angled triangle in which ∠A = 90° and AB = AC. Find ∠B and ∠C.