Types of Quadrilaterals
Parallelogram
Definition: Parallelogram is a quadrilateral where both pairs of the opposite sides are parallel to each other.
In the figure shown,
Note that AC is a transversal to . Similarly, AC is also a transversal to
.
Properties of a Parallelogram (please refer to the above diagram):
1) Opposite sides are equal and parallel to each other.
2) Opposite angles are equal
3) Adjacent angles are supplementary (consecutive interior angles)
4) The diagonals (AC and BD) bisect each other
5) Each diagonal bisect the parallelogram in two congruent triangles
AC bisects the parallelogram ABCD in such that in
BD bisects the parallelogram ABCD in such that in
Theorem: In a parallelogram,
Opposite sides are equal
Opposite angles are equal
Each diagonal bisects the parallelogram
Proof:
Given: Parallelogram such that
To Prove:
i.e. area of
is equal to area of
Similarly, i.e. area of
is equal to area of
Proof:
(alternate angles because
is a transversal)
(alternate angles because
is a transversal)
DB is common
Therefore, [A.S.A axiom]
i) Since
ii) Since
Hence,
Since
Area of (since congruent triangles are equal in area)
Similarly, if we were to join AC, we could use the same logic to prove .
Rhombus
Definition: A parallelogram where all four sides are equal is called a Rhombus.
Properties of a Rhombus:
1) Opposite sides are parallel to each other.
2) All sides are equal
3) Diagonals AC and BD bisect each other at right angles
4) Diagonal (i.e.
). Similarly, diagonal
Similarly, .
Theorem: The diagonals of a Rhombus bisect each other at right angles.
Given: Rhombus are diagonals.
To Prove:
Proof:
(alternate angles)
(alternate angles)
(Side of a Rhombus)
Therefore (A.S.A Axiom)
Hence
Again in
(Side of a Rhombus)
(Proved above) And
is common
Hence (S.S.S axiom)
Therefore
Now
Or