**3.5 SOME SPECIAL PARALLELOGRAMS:**

**Rhombus.**

If a parallelogram has all the four sides equal then it is called a Rhombus.

Here, AB = BC = CD = AD.P

A rhombus has all the properties of a parallelogram and a kite.

The special property is that its diagonals are perpendicular bisectors of each other.

Hence, AE = EC and DE = EB.

It makes an angle of 90° at the point of intersection of the diagonals.

**Rectangle:**

**A parallelogram with equal angles and equal opposite sides is called Rectangle.**

Here, AB = DC and AD = BC.

**Properties of a Rectangle:**

Opposite sides are parallel and equal.

Opposite angles are equal.

Adjacent angles make a pair of supplementary angles.

Diagonals are of equal length.

Diagonals bisect each other.

All the four angles are of 90°.

**Square:**

**A rectangle with all the four equal sides is called a Square.**

Here, AB = BC = CD = AD.

**Properties of a square:**

- All four sides are equal.
- Opposite sides are parallel.
- Diagonals are of equal length.
- Diagonals are perpendicular bisector to each other.
- All the four angles are of 90°.

**Example: **Find the perimeter of the parallelogram ABCD .

** Solution: **In a parallelogram, the opposite sides have same length.

∴ AB = CD = 15 cm and BC = AD = 16 cm

So, Perimeter = AB + BC + CD + AD

= 15 cm + 6 cm + 15 cm + 6 cm = 42 cm

**Example 2: ** In PQRS, is a parallelogram. Find the values *x*, *y *and *z*.

**Solution: **R is opposite to P.

So, *x * = 85° ( opposite angles property )

* y * = 95° ( measure of angle corresponding to ∠ *x* )

* z* = 85° ( since∠ *y*, ∠ *z *is a linear pair )

We to find the adjacent angles of a parallelogram.

In parallelogram ABCD,

∠A and ∠D are supplementary since

AD, BC and BA is a transversal, thus making ∠A and ∠B interior opposite angles.

**Property: ** The adjacent angles in a parallelogram are supplementary.

**Example 3****: **** Prove that any two adjacent angles of a parallelogram are supplementary.**

*Solution***: ** Let ABCD be a parallelogram

Then, AD ∥ BC and AB is a transversal.

Therefore, ∠A + ∠B = 180° [ Since, sum of the interior angles on the same side of the transversal is 180°]

Similarly, ∠B + ∠C = 180°, ∠C + ∠D = 180° and ∠ D + ∠A = 180°. Thus, the sum of any two adjacent angles of a parallelogram is 180°. Hence, any two adjacent angles of a parallelogram are supplementary.

### EXERCISE 3.4

**State whether True or False.**

( a ) All rectangles are squares.

Ans**: ** False. Because, all square are rectangles but all rectangles are not square.

( b ) All rhombuses are parallelograms. ( Ans ) True

( c ) All squares are rhombuses and also rectangles. ( Ans ) True

( d ) All squares are not parallelograms*.*

Ans**: ** False. Because, all squares are parallelograms as opposite sides are parallel and opposite angles are equal.

( e ) All kites are rhombuses.

Ans**: ** False. Because, lengths of the sides of a kite are not of same length.

( f ) All rhombuses are kites. ( Ans ) True

( g ) All parallelograms are trapeziums. ( Ans ) True

( h ) All squares are trapeziums. ( Ans ) True

**Identify all the quadrilaterals that have.**

**( a ) four sides of equal length ( b ) four right angles**

*Solution***: **

( a ) Rhombus and square have all four sides of equal length.

( b ) Square and rectangle have four right angles.

**Explain how a square is.**

**( i ) a quadrilateral ( ii ) a parallelogram **

**( iii ) a rhombus ( iv ) a rectangle**

*Solution*:

( i ) Square is a quadrilateral because it has four sides.

( ii ) Square is a parallelogram because it’s opposite sides are parallel and opposite angles

are equal.

( iii ) Square is a rhombus because all the four sides are of equal length and diagonals bisect at right angles.

( iv ) Square is a rectangle because each interior angle, of the square, is 90°

**Name the quadrilaterals whose diagonals.**

**( i ) bisect each other ( ii ) are perpendicular bisectors of each other ( iii ) are equal **

*Solution***: **

( i ) Parallelogram, Rhombus, Square and Rectangle

( ii ) Rhombus and Square

( iii ) Rectangle and Square

**Explain why a rectangle is a convex quadrilateral.**

*Solution***: **Rectangle is a convex quadrilateral because both of its diagonals lie inside the rectangle.

**ABC is a right – angled triangle and O is the mid – point of the side opposite to the right angle. Explain why O is equidistant from A, B and C.**

*Solution***:**AD and DC are drawn so that AD || BC and AB || DCAD = BC and AB = DCABCD is a rectangle as opposite sides are equal and parallel to each other and all the

interior angles are of 90°.

In a rectangle, diagonals are of equal length and also bisects each other.

Hence, AO = OC = BO = OD.Thus, O is equidistant from A, B and C.