Contents
Trapezium:
A quadrilateral having exactly one pair of parallel sides, is called a trapezium.
ABCD is a trapezium in which AB || DC.
Isosceles Trapezium:
A trapezium is said to be an isosceles trapezium, if its non-parallel sides are equal.
Thus, a quadrilateral ABCD is an isosceles trapezium, if AB || DC and AC = BD.
The diagonals of an isosceles trapezium are always equal.
Parallelogram:
A quadrilateral is a parallelogram if its both pairs of opposite sides are parallel.
Quadrilateral ABCD is a parallelogram, because AB || DC and AD || BC.
Parallelogram Properties:
1) The opposite sides of a parallelogram are equal and parallel.
2) The opposite angles of a parallelogram are equal.
3) The diagonals of a parallelogram bisect each other.
Thus, in a parallelogram ABCD, we have:
1) AB = DC, AD = BC and AB || DC, AD || BC.
2) \(\angle{BAD} = \angle{BCD}\) and \(\angle{ABC} = \angle{ADC}\).
3) If the diagonals AC and BD intersect at O, then OA = OC and OB = OD.
Rhombus:
A parallelogram having all sides equal is called a rhombus.
Thus, a parallelogram ABCD is a rhombus if AB = AD.
In other words, a quadrilateral ABCD is a rhombus if AB || DC, AD || BC and AB = BC = CD = DA.
ABCD is a rhombus, since ABCD is a parallelogram in which AB = AD.
Rhombus Properties:
1) The opposite sides of a rhombus are parallel.
2) All sides of a rhombus are equal.
3) The opposite angles of a rhombus are equal.
4) The diagonals of a rhombus bisect each other at right angles.
Thus, in a rhombus ABCD, we have:
1) AB || DC and AD || BC.
2) AB = BC = CD = DA.
3) \(\angle{DAB} = \angle{BCD}\) and \(\angle{ABC} = \angle{CDA}\).
4) Let the diagonals AC and BD intersect at O. Then, OA = OC. OB =O D and \(\angle{AOB} = \angle{COD}\) and \(\angle{BOC} = \angle{AOD} = 1 right angle\).
Rectangle:
A parallelogram whose each angle is a right angle, is called a rectangle.
ABCD is a rectangle in which AB || CD, AD || BC and \(\angle{A} = \angle{B} = \angle{C} = \angle{D}\) = 90°.
Rectangle Properties:
1) Opposite sides of a rectangle are equal and parallel.
2) Each angle of a rectangle is 90°.
3) Diagonals of a rectangle are equal.
Thus, In a rectangle ABCD, we have:
1) AB = DC, AD = BC and AB || DC, AD || BC.
2) \(\angle{A} = \angle{B}\) and \(\angle{C} = \angle{D} = 1 right angle\).
3) Diagonal AC = diagonal BD.
Square:
A square is a rectangle with a pair of adjacent sides equal.
In other words, a parallelogram having all sides equal and each angle equal to a right angle, is called a square.
ABCD is a square in which AB || DC, AD || BC, AB = BC = CD = DA and \(\angle{A} = \angle{B} = \angle{C} = \angle{D}\) = 90°.
Square Properties:
1) The sides of a square are all equal.
2) Each angle of a square is 90°.
3) The diagonals of a square are equal and bisect each other at right angles.
Thus, in a square ABCD, we have:
1) AB = BC = CD = DA.
2) \(\angle{A} = \angle{B}\) and \(\angle{C} = \angle{D}\) = 90°.
3) Diagonal AC = Diagonal BD.
Kite:
A quadrilateral is a kite if it has two pairs of equal adjacent sides and unequal opposite sides.
Thus, a quadrilateral ABCD is a kite, if AB = AD, BC = CD but \({AD}\neq{BC}\) and \({AB}\neq{CD}\).