Special Quadrilaterals

A quadrilateral is a polygon with four sides and four vertices. While a general quadrilateral has no guaranteed equal sides or angles, special quadrilaterals follow specific rules regarding their sides, angles, and diagonals.

The Parallelogram Family

A parallelogram is a quadrilateral with two pairs of parallel opposite sides. Rectangles, rhombuses, and squares are all special types of parallelograms.

1. Rectangle

A rectangle is a parallelogram with four right angles ($90^\circ$).

• Key Property: The diagonals of a rectangle are equal in length ($AC = BD$) and bisect each other.

2. Rhombus

A rhombus is a parallelogram with four equal sides.

• Key Property: The diagonals of a rhombus are perpendicular to each other ($AC \perp BD$) and bisect the interior angles.

3. Square

A square is a regular quadrilateral, meaning it has four equal sides and four right angles. It is both a rectangle and a rhombus.

• Key Property: The diagonals are equal in length, bisect each other at exactly $90^\circ$, and bisect the opposite angles.

Other Special Quadrilaterals

1. Trapezoid

A trapezoid (or trapezium) is a quadrilateral with exactly one pair of parallel sides, called the bases.

• Isosceles Trapezoid: A trapezoid where the non-parallel sides (legs) are equal in length. In an isosceles trapezoid, the base angles are equal, and the diagonals are equal in length.

2. Kite

A kite is a quadrilateral with two distinct pairs of adjacent equal sides.

• Key Properties: The diagonals of a kite are perpendicular to each other. One diagonal is the perpendicular bisector of the other, and exactly one pair of opposite angles is equal.

Example Problems

Example 1: Finding the side length of a rhombus

Problem: In rhombus $ABCD$, the diagonals are $10$ and $24$. Find the side length.

Solution: The diagonals of a rhombus bisect each other at right angles ($90^\circ$). This intersection divides the rhombus into four identical right-angled triangles.

The legs of each right triangle are half the lengths of the diagonals: $a = \frac{10}{2} = 5$ $b = \frac{24}{2} = 12$

Use the Pythagorean theorem ($a^2 + b^2 = c^2$) to find the hypotenuse, which is the side length ($s$) of the rhombus: $s^2 = 5^2 + 12^2$ $s^2 = 25 + 144$ $s^2 = 169$ $s = 13$

Answer: The side length of the rhombus is $13$.

Example 2: Proving a rectangle

Problem: Prove that if a parallelogram has one right angle, it is a rectangle.

Proof: Let the parallelogram be $ABCD$ with $\angle A = 90^\circ$.

1. In any parallelogram, consecutive angles are supplementary (they add up to $180^\circ$). Therefore, $\angle B = 180^\circ - \angle A = 180^\circ - 90^\circ = 90^\circ$.
2. In a parallelogram, opposite angles are equal. Therefore, $\angle C = \angle A = 90^\circ$ and $\angle D = \angle B = 90^\circ$.
3. Since all four angles ($\angle A, \angle B, \angle C, \angle D$) are $90^\circ$, the parallelogram meets the definition of a rectangle.

Conclusion: A parallelogram with at least one right angle must be a rectangle.