3.5 SOME SPECIAL PARALLELOGRAMS:
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.
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°.
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.
- 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
( 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
( 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
( 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
( 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 = 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.