ABCD is a parallelogram AP and CQ are perpendiculars drawn from vertices A and C on diagonal BD (see figure)..Show that : i) `DeltaAPB = DeltaCQD`.

ABCD is a parallelogram and AP and CQ are perpendicular from vertices A and C on diagonal BD (see figure) show that : /_\APB ~= /_\CQD

ABC is a triangle in which altitudes BD and CE to sides AC and AB are equal (see figure) . Show that (i) DeltaABD ~= DeltaACE (ii) AB = AC i.e., ABC is an isosceles triangle.

In right triangle ABC, right angle is at C, M is the mid-point of hypotenuse AB. C is joined to M and produced to a point D such that DM = CM. Point D is joined to point B (see figure). Show that : (i) DeltaAMC ~= DeltaBMD (ii) /_DBC is a right angle (iii) Delta DBC ~= DeltaACB (iv) CM = 1/2 AB .

AB and CD are respectively the smallest and longest sides of aquadrilateral ABCD (see adjacent figure). Show that /_A > /_C and /_ B > /_ D.

ABCD is a parallelogram. AC and BD are the diagonals intersect at O. P and Q are the points of tri section of the diagonal BD. Prove that CQ" ||" AP and also AC bisects PQ (see figure).

In Delta ABC , if AD , BE , CF are the perpendiculars drawn from the vertices A, B, C to the opposite sides, shot that (1)/( AD) + ( 1)/(BE) + (1)/(CF) = (1)/ ( r ) and (ii) AD . BE. CF = ((abc) ^(2))/( 8R^(3)) =( 8 Delta ^(3))/( abc)

AB and CD are respectively the small est and longest sides of a quadrilateral ABCD (see adjacent figure).Show that /_Agt/_C /_Bgt/_D .

P is a point equidistant from two lines l and m intersecting at point A (see figure). Show that the line AP bisects the angle between them.