M l aggarwal understanding icse mathematics class 9 Solutions for Chapter 13 rectilinear figures

If two angles of a quadrilateral are 40° and 110° and the other two are in the ratio 3 : 4, find

these angles.

If the angles of a quadrilateral, taken in order, are in the ratio 1: 2 : 3: 4, prove that it is a

trapezium.

(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.

Prove that:

(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

other.

If P and Q are points of trisection of the diagonal BD of a parallelogram ABCD, prove that CQ

|| AP.

.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.

1. In a parallelogram ABCD, the bisector of ∠A meets DC in E and AB = 2 AD. Prove that

(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.

ABCD is a rhombus in which ∠A = 60°. Find the ratio AC : BD.

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.

Construct a trapezium ABCD in which AD || BC, ∠B= 60°, AB = 5 cm. BC = 6.2 cm and CD =

4.8 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

ABCD.

Using ruler and compasses, construct a rectangle ABCD with AB = 5cm and AD = 3 cm.

Using ruler and compasses only construct A rhombus ABCD given that AB 5cm, AC = 6cm

measure ∠BAD.

Construct a rhombus PQRS whose diagonals PR and QS are 8cip and 6cm respectively.

Construct a rhombus ABCD of side 4.6 cm and ∠BCD = 135°, by using a ruler and compasses

only.

Construct a trapezium ABCD when one of parallel sides AB = 4.8 cm, height = 2.6cm, BC = 3.1

cm and AD = 3.6 cm.