A quadrilateral is a closed plane figure bounded by four line segments. For example, the figure ABCD shown below is a quadrilateral.

A line segment drawn from one vertex of a quadrilateral to the opposite vertex is called a diagonal of the quadrilateral.

For example, AC is a diagonal of quadrilateral ABCD, and so is BD.

There are six types of special quadrilaterals as shown in the diagram below.

**Kite : **

**Properties of Kite : **

- Two pairs of consecutive congruent sides, but opposite sides are not congruent.
- Exactly one pair of opposite angles are equal.
- Diagonals intersect at right angles.
- The longest diagonal bisects the shortest diagonal into two equal parts.

**Parallelogram : **

**Properties of Parallelogram : **

- Opposite sides are parallel and equal.
- Opposite angles are equal.
- Diagonals bisect each other.

**Trapezoid :**

**Properties of Trapezoid : **

- A trapezium has exactly one pair of opposite sides parallel.
- A regular trapezium has non-parallel sides equal and its base angles are equal as shown in the diagram above.

**Rhombus :**

**Properties of Rhombus : **

- All sides are equal and opposite sides are parallel.
- Opposite angles are equal.
- The diagonals bisect each other at right angles.

**Rectangle : **

**Properties of Rectangle : **

- Opposite sides are parallel and equal.
- All angles are 90º.
- The diagonals bisect each other.

**Square : **

**Properties of Square : **

- Opposite sides are parallel and all sides are equal.
- All angles are 90º.
- Diagonals bisect each other at right angles.

**Question 1 : **

"A quadrilateral has at least one pair of opposite sides congruent"

What type of special quadrilateral can meet the above condition ?

**Answer : **

There are many possibilities.

**Parallelogram : **

Opposite sides are congruent.

**Rhombus :**

All sides are congruent.

**Rectangle : **

Opposite sides are congruent.

**Square : **

All sides are congruent.

**Isosceles Trapezoid : **

Legs are congruent.

**Question 2 : **

When we join the midpoints of the sides of any quadrilateral, what type of special quadrilateral formed ? Explain your answer.

**Answer : **

Let E, F, G and H be the midpoints of the sides of the quadrilateral ABCD shown below.

If we draw AC, the Midsegment Theorem for Triangles says

FG || AC and EH || AC ----> FG || EH

Similar reasoning shows that

EF || HG

So, by definition, EFGH is a parallelogram.

Hence, when the midpoints of the sides of any quadrilateral are joined, the type of special quadrilateral formed is parallelogram.

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