If two angles of a quadrilateral are 40° and 110° and the other two are in the ratio 3 : 4, find
If the angles of a quadrilateral, taken in order, are in the ratio 1: 2 : 3: 4, prove that it is a
(a) In figure (1) given below, ABCD is a parallelogram in which ∠DAB = 70°, ∠DBC = 80°.
Calculate angles CDB and ADB.
(b) In figure (2) given below, ABCD is a parallelogram. Find the angles of the AAOD.
(c) In figure (3) given below, ABCD is a rhombus. Find the value of x.
Given, ABCD is a rectangle
We know that the diagonals of a rectangle are the same
and bisect each other
So, we have
AP = BP
(i) Prove that each angle of a rectangle is 90°.
(ii) If the angle of a quadrilateral is equal, prove that it is a rectangle.
(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.
(i) Prove that bisectors of any two adjacent angles of a parallelogram are at right angles.
(ii) Prove that bisectors of any two opposite angles of a parallelogram are parallel.
(iii) If the diagonals of a quadrilateral are equal and bisect each other at right angles, then prove
that it is a square.
(a) In figure (1) given below, ABCD is a parallelogram and X is the mid-point of BC. The line AX
produced meets DC produced at Q. The parallelogram ABPQ is completed.
(i) the triangles ABX and QCX are congruent;
(ii)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
If P and Q are points of trisection of the diagonal BD of a parallelogram ABCD, prove that CQ
.A transversal cuts two parallel lines at A and B. The two interior angles at A are bisected and
so are the two interior angles at B; the four bisectors form a quadrilateral ABCD. Prove that
(i) ABCD is a rectangle.
(ii) CD is parallel to the original parallel lines.
(i) BE bisects ∠B
(ii) ∠AEB = a right angle.
In a parallelogram ABCD, the bisector of ∠A meets DC in E and AB = 2 AD. Prove that:
(i) BE bisects ∠B
(ii) ∠AEB is a right angle
ABCD is a square and the diagonals intersect at O. If P is a point on AB such that AO =AP,
prove that 3 ∠POB = ∠AOP.
ABCD is a square. E, F, G and H are points on the sides AB, BC, CD and DA respectively such
that AE = BF = CG = DH. Prove that EFGH is a square.
Using ruler and compasses only, construct the quadrilateral ABCD in which ∠ BAD = 45°, AD =
AB = 6cm, BC = 3.6cm, CD = 5cm. Measure ∠ BCD.
Using ruler and compasses only, construct the quadrilateral ABCD given that AB = 5 cm, BC =
2.5 cm, CD = 6 cm, ∠BAD = 90° and the diagonal AC = 5.5 cm.
Using ruler and compasses only, construct a parallelogram ABCD with AB = 5.1 cm, BC = 7 cm
and ∠ABC = 75°.
Using ruler and compasses, construct a parallelogram ABCD give that AB = 4 cm, AC = 10 cm,
BD = 6 cm. Measure BC.
Using ruler and compasses only, construct a parallelogram ABCD such that BC = 4 cm,
diagonal AC = 8.6 cm and diagonal BD = 4.4 cm. Measure the side AB.
Using ruler and compasses only, construct a parallelogram ABCD with AB = 6 cm, altitude =
3.5 cm and side BC = 4 cm. Measure the acute angles of the parallelogram.
The perpendicular distances between the pairs of opposite sides of a parallelogram ABCD are
3 cm and 4 cm and one of its angles measures 60°. Using a ruler and compasses only, construct
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