Question

AB is a line-segment. P and Q are points on opposite sides of AB such that each of them is equidistant from the points A and B (see Fig. 7.37). Show that the line PQ is the perpendicular bisector of AB.

△ ABC and △ DBC are two isosceles triangles on the same base BC and vertices A and D are on the same side of BC (see Fig. 7.39). If AD is extended to intersect BC at P, show that (i) △ ABD ≅ △ ACD (ii) △ ABP ≅ △ ACP (iii) AP bisects ∠ A as well as ∠ D. (iv) AP is the perpendicular bisector of BC.

Two sides AB and BC and median AM of one triangle ABC are respectively equal to sides PQ and QR and median PN of △ PQR (see Fig. 7.40). Show that: (i) △ ABM ≅ △ PQN (ii) △ ABC ≅ △ PQR

ABCD is a quadrilateral in which AB = BC and AD = CD. Show that BD bisects both the angles ABC and ADC.

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.