3.5    SOME SPECIAL PARALLELOGRAMS:

  1. 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.

  1. 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°.

  1. 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

  1. 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

  1. 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.

  1. 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°

  1. 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

  1. Explain why a rectangle is a convex quadrilateral.

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

  1. 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   =   DC

    ABCD 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.