ICSE Class 8 Maths Geometry Chapter 04 Quadrilaterals

Quadrilaterals

Quadrilateral

A quadrilateral is a polygon with four sides. The quadrilateral ABCD shown in the figure has:

Four sides AB, BC, CD and DA

Four vertices A, B, C and D

Four angles \(\angle A\) (or \(\angle DAB\)), \(\angle B\) (or \(\angle ABC\)), \(\angle C\) (or \(\angle BCD\)) and \(\angle D\) (or \(\angle CDA\))

Two diagonals AC and BD.

Teacher's Note

Quadrilaterals are everywhere around us - from the tiles on our floors to the windows in our homes, helping us understand the geometry of everyday spaces.

Parallelogram

A quadrilateral is called a parallelogram if its opposite sides are parallel. In the figure, ABCD is a parallelogram, in which AB is parallel to DC and AD is parallel to BC. A parallelogram has some special properties, which we will now study.

Theorem 1

(i) The opposite sides of a parallelogram are equal.

(ii) The opposite angles of a parallelogram are equal.

(iii) Each diagonal bisects a parallelogram into two congruent triangles.

Given ABCD is a parallelogram in which AB \(\parallel\) DC and AD \(\parallel\) BC.

To Prove (i) AB = DC and BC = AD,

(ii) \(\angle A = \angle C\) and \(\angle B = \angle D\),

(iii) \(\triangle ABC \cong \triangle CDA\) and \(\triangle ABD \cong \triangle CDB\).

Construction Join the points A and C.

Proof In \(\triangle ABC\) and \(\triangle CDA\),

\(\angle 1 = \angle 2\) (\(\because\) AB \(\parallel\) DC, alternate angles are equal),

AC = AC (common),

and \(\angle 3 = \angle 4\) (\(\because\) BC \(\parallel\) AD, alternate angles are equal).

\(\therefore \triangle ABC \cong \triangle CDA\) (A-S-A condition of congruency).

So, the corresponding parts of the triangles are equal.

\(\therefore\) AB = DC and BC = AD. (Proved)

Also, \(\angle B = \angle D\).

We have \(\angle 1 = \angle 2\) and \(\angle 4 = \angle 3\). So, \(\angle 1 + \angle 4 = \angle 2 + \angle 3 \Rightarrow \angle A = \angle C\).

Teacher's Note

Understanding parallelograms helps us design stable structures in construction and engineering, from bridges to building frameworks that distribute forces evenly.

So, \(\angle A = \angle C\) and \(\angle B = \angle D\). (Proved)

Now, \(\triangle ABC \cong \triangle CDA\) (proved already). Similarly, \(\triangle ABD \cong \triangle CDB\).

Hence, each diagonal bisects the parallelogram into two congruent parts. (Proved)

Theorem 2

The diagonals of a parallelogram bisect each other.

Given ABCD is a parallelogram in which AB \(\parallel\) DC, AD \(\parallel\) BC and the diagonals AC and BD intersect at the point O.

To prove OA = OC and OB = OD.

Proof In \(\triangle OAB\) and \(\triangle OCD\),

\(\angle OAB = \angle OCD\) (AB \(\parallel\) DC and alternate angles are equal)

AB = DC (Opposite sides of a parallelogram are equal)

and \(\angle OBA = \angle ODC\) (AB \(\parallel\) DC and alternate angles are equal).

\(\therefore \triangle OAB \cong \triangle OCD\) (A-S-A condition of congruency).

So, the corresponding sides of \(\triangle OAB\) and \(\triangle OCD\) are equal.

\(\therefore\) OA = OC and OB = OD. (Proved)

Theorem 3

If a pair of opposite sides of a quadrilateral are equal and parallel, the quadrilateral is a parallelogram.

Given ABCD is a quadrilateral in which AB = DC and AB \(\parallel\) DC.

To prove ABCD is a parallelogram.

Construction Join the points B and D.

Proof In \(\triangle ABD\) and \(\triangle CDB\),

AB = DC (Given)

\(\angle ABD = \angle CDB\) (\(\because\) AB \(\parallel\) DC, alternate angles are equal)

and BD = DB (Common side).

\(\therefore \triangle ABD \cong \triangle CDB\) (S-A-S condition of congruency).

So, the corresponding parts of these triangles are equal.

\(\therefore\) \(\angle ADB = \angle CBD\), but these are alternate angles. So, AD \(\parallel\) BC.

Thus, AB \(\parallel\) DC and AD \(\parallel\) BC. Hence, ABCD is a parallelogram. (Proved)

Thus, the properties of a parallelogram are:

Its opposite sides are equal.

Its opposite angles are equal.

Its diagonals bisect each other.

Each of its diagonals divides it into two congruent triangles.

Its adjacent angles are supplementary because they are co-interior angles and the opposite sides are parallel.

Rectangle

A parallelogram is called a rectangle if one of its angles is a right angle. In the figure, ABCD is a parallelogram in which \(\angle A = 90°\). Thus, ABCD is a rectangle. A rectangle has all the properties of a parallelogram. In addition, it has the following special properties.

Teacher's Note

Rectangles are fundamental to architecture and design, forming the basis of most building layouts, windows, and doors that make our spaces functional and organized.

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