1. Parallelogram
Definition
A parallelogram is a quadrilateral in which opposite sides are parallel.
Properties of a Parallelogram
Remember these important properties:
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Opposite sides are parallel.
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Opposite sides are equal.
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Adjacent angles add up to 180°.
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Opposite angles are equal.
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Diagonals bisect each other.
Angle Property
If one angle is known:
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The opposite angle is equal to it.
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Each adjacent angle is supplementary to it.
Therefore:
Adjacent angles = 180°
Opposite angles = equal
Diagonal Property
The diagonals of a parallelogram bisect each other.
If diagonals AC and BD intersect at O:
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AO = OC
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BO = OD
Important Relationship
A rectangle is a special type of parallelogram because its opposite sides are parallel.
A square is also a special type of parallelogram.
So:
Square ⟶ Rectangle ⟶ Parallelogram
Every square is a rectangle and every rectangle is a parallelogram.
But the reverse is not necessarily true.
2. Rhombus
Definition
A rhombus is a quadrilateral in which all four sides have the same length.
Properties of a Rhombus
Remember these four properties:
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All four sides are equal.
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Opposite sides are parallel.
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Adjacent angles add up to 180° and opposite angles are equal.
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Diagonals bisect each other at right angles.
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Diagonals bisect the angles of the rhombus.
Diagonal Properties
The diagonals of a rhombus:
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Bisect each other.
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Intersect at 90°.
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Bisect the angles of the rhombus.
So, if diagonals intersect at O:
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Each diagonal divides the other into two equal parts.
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The angle formed at their intersection is 90°.
Important Relationship
A square is a special type of rhombus because all four sides of a square are equal.
Also:
Square ⟶ Rhombus
But every rhombus is not necessarily a square.
3. Kite
Definition
A kite is a quadrilateral with two non-overlapping adjacent pairs of sides having the same length.
In other words, the equal sides occur in adjacent pairs.
For example:
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One pair of adjacent sides is equal.
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The other pair of adjacent sides is also equal.
Important Properties of a Kite
The key diagonal properties are:
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One diagonal bisects the other diagonal.
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One diagonal bisects a pair of opposite angles.
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The diagonals are perpendicular to each other.
Remember
A kite does not have to have all four sides equal.
Therefore:
Every rhombus is a kite, but every kite is not a rhombus.
A rhombus has all four sides equal, whereas a general kite has two pairs of adjacent equal sides.
4. Trapezium
Definition
A trapezium is a quadrilateral having at least one pair of parallel opposite sides.
The parallel sides are called the parallel sides or bases.
The other two sides are the non-parallel sides.
Isosceles Trapezium
An isosceles trapezium is a trapezium in which the non-parallel sides are equal.
Properties of an Isosceles Trapezium
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The non-parallel sides are equal.
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The angles at each base are equal.
Therefore:
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One pair of opposite angles is equal.
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The other pair of opposite angles is also equal.
Angle Idea
Since the parallel sides are parallel, the interior angles on the same side of a non-parallel side are supplementary.
So the relevant pairs of angles add up to: 180°
