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