Please help sum11 (iii)

Please help sum11 (iii) Prove that angle of a rectangle iS 900. If the angle of a quadrilateral are equal, prove that it iS a (ii0 If the diagonals of a rhombus are equal, prove that it is a square. (iv) Prove that every diagonal of a rhombus bisects the angles at the vertices. ABCD is parallelogram. If the diagonal AC bisects LA, then prove (1) AC bisects Z C (ii) ABCD is a rhombus (iii) AC -L BD. (i) Prove that bisectors of any two adjacent angles of a parallelogram are at right angles. (i) prove that bisectors of any two opposite angles of a parallelogram are parallel. (iii) If the diagonals ofa quadrilateral are equal and bisect each other at right angles, then prove that it is a square. 12 (D If A BCD is a rectangle in which the diagonal BD bisects Z B, then show th at ABCD is a square. (ii) Show that if the diagonals of a quadrilateral are equal and bisect each other at right angles, then it is a square. g P and Q are points on opposite sides AD and BC of a parallelogram ABCD such that PQ passes through the point of intersection O of its diagonals AC and BD. Show that PQ is bisected at O. Hint. Show that AOAP AOCQ. (a) In figure ( 1 ) given below, ABCD is a parallelogram and X is mid -point of BC. The line AX produced meets DC produced at Q. The parallelogram ABPQ is completed. Prove that (i) the triangles ABX and QCX are congruent. DC = CQ = QI' (b) In figure (2) given below, points P and Q have been taken on opposite sides AB and CD respectively of a parallelogram ABCD such that AP = QQ. Show that AC and PQ bisect each other. Q c o D 15' ABCD is a square. A is joined to a point P on BC and D is joined to a point Q on AB. If AP = DQ prove that AP and DQ are perpendicular to each other. Hint. DAQ— LBAP = ZADQ. But LBAD = 900 ZPAD + ZADQ = 900. If P and Q are points of trisection of the diagonal BD Of a parallelogram ABCD, prove that CQ II AP. Hint. AABP ACDQ. Ü• A transversal cuts two parallel lines at A and B. The two interior angles at A bisected and so are the two interior angles at B; the four bisectors form a quadrilateral ACBD