ICSE Class 9 Maths Chapter 15 Quadrilaterals

15. Quadrilaterals

Points To Remember

1. Quadrilateral. A closed four sided figure is called a quadrilateral.

(i) It has four sides, four vertices, four angles and two diagonals.

(ii) Sum of its four angles = 360° i.e. \(\angle A + \angle B + \angle C + \angle D = 360°\).

2. Types Of Quadrilaterals

1. Parallelogram. A quadrilateral in which opposite sides are parallel, is called a parallelogram.

In the given figure, ABCD is a quadrilateral in which AB \(\parallel\) DC and AD \(\parallel\) BC.

Therefore ABCD is a parallelogram.

Its opposites sides are equal i.e. AB = CD and AD = BC.

2. Rhombus. A parallelogram having all sides equal, is called a rhombus.

In the given figure, ABCD is a parallelogram in which AB = BC = CD = DA.

Therefore ABCD is a rhombus.

3. Rectangle. A parallelogram each of whose angles measures 90°, is called a rectangle.

In the given figure, ABCD is a rectangle.

Its opposite sides are equal.

4. Square. A rectangle having all sides equal, is called a square.

In the given figure, PQRS is a square, in which PQ = QR = RS = SP.

Its each angle is of 90°.

Teacher's Note

Quadrilaterals form the foundation of architectural design, from building floors to window panes, helping us understand how spaces are organized and measured.

5. Trapezium.

A quadrilateral having two parallel opposite sides and two non-parallel opposite sides is called a trapezium.

In the given figure, ABCD is a quadrilateral in which AB \(\parallel\) DC and AD is not parallel to BC.

Therefore ABCD is a trapezium.

If the non-parallel sides of a trapezium are equal, then it is called an isosceles trapezium.

6. Kite. A quadrilateral in which two pairs of adjacent sides are equal, is known as a kite.

In the given figure, ABCD is a quadrilateral in which AB = AD and CB = CD.

Therefore ABCD is a kite.

Results on Parallelogram

Theorem 1. In a parallelogram:

(i) the opposite sides are equal;

(ii) the opposite angles are equal;

(iii) each diagonal bisects the parallelogram.

Theorem 2. The diagonals of a parallelogram bisect each other.

Theorem 3. If a pair of opposite sides of a quadrilateral are parallel and equal, then it is a parallelogram.

Teacher's Note

Understanding parallelograms helps in recognizing patterns in everyday objects like tiles, books, and doors, which maintain their shapes through parallel alignment.

Exercise 15 (A)

Q. 1. In the given figure, ABCD is a parallelogram in which \(\angle A = 70°\). Calculate \(\angle B\), \(\angle C\) and \(\angle D\).

Sol. \(\therefore\) ABCD is a parallelogram.

\(\therefore \angle A = \angle C\) and \(\angle B = \angle D\)

\(\therefore \angle C = \angle A = 70°\)

But \(\angle A + \angle B = 180°\) (Co. interior angles)

\(\Rightarrow 70° + \angle B = 180°\)

\(\Rightarrow \angle B = 180° - 70° = 110°\)

But \(\angle D = \angle B\)

\(\therefore \angle D = 110°\)

Hence \(\angle B = 110°\), \(\angle C = 70°\) and \(\angle D = 110°\) Ans.

Q. 2. In the given figure, ABCD is a parallelogram. Side DC is produced to E and \(\angle BCE = 105°\).

Calculate \(\angle A\), \(\angle B\), \(\angle C\) and \(\angle D\).

Sol. ABCD is a parallelogram. Side DC is produced to E and \(\angle BCE = 105°\)

But \(\angle BCE + \angle BCD = 180°\) (Linear pair)

\(\Rightarrow 105° + \angle BCD = 180°\)

\(\Rightarrow \angle BCD = 180° - 105° = 75°\)

or \(\angle C = 75°\)

But \(\angle A = \angle C\) (Opposite angles of a parallelogram)

\(\therefore \angle A = 75°\)

\(\therefore\) AB \(\parallel\) CD

\(\therefore \angle BCE = \angle CBA\) (Alternate angles)

\(\Rightarrow \angle CBA = 105°\) or \(\angle B = 105°\)

But \(\angle D = \angle B\) (Opposite angles of a parallelogram)

\(\therefore \angle D = 105°\)

Hence \(\angle A = 75°\), \(\angle B = 105°\), \(\angle C = 75°\), \(\angle D = 105°\) Ans.

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