Please help sum 13 Prove angle of a rectangle is 900.
that it is a
(iO If the angle of a quadrilateral are equal, prove
(iii) 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 that:
(O AC bisects Z C (ii) A BCD is a rhombus (iii) AC -L BD.
(i) Prove that bisectors of any two adjacent angles of a parallelogram are at right
(ii) Prove that bisectors of any two opposite angles of a parallelogram are parallel.
(ih) If the diagonals of a quadrilateral are equal and bisect each Other at right angles,
then prove that it is a square.
(1) If A BCD is a rectangle in which the diagonal BD bisects Z B, then show that
ABCD is a square.
(if) Show that if the diagonals of a quadrilateral are equal and bisect each Other at
right angles, then it is a square.
pand 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.
14 (a) In fi gure (l ) gi ven below. ABCD is a parallelogram and X is mi d-point Of BC.
The line AX produced meets DC produced at Q. The parallelogram ABPQ is
completed. Prove that
(1) the triangles ABX and QCX are congruent.
(h) DC = CQ=QP
(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 = CQ. Show that
AC and PQ bisect each other.
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. AABP DAQ ZBAP = ZADQ. But z BAD 900
ZPAD + ZADQ 90'.
If P and Q are points of trisection of the diagonal BD Of a parallelogram ABCD, prove
that CQ II AP.
Hint. ABP = ACDQ.
A transversal cuts two parallel lines at A and B. The two interior angles at Aare bisected
and so are the two interior angles at B; the four bisectors form a quadrilateral ACBD