# 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