Classify quadrilaterals by their properties

Detailed Explanation: Classify quadrilaterals by their properties

A quadrilateral is classified by its properties, not by how it looks. Check its sides, angles, and parallel lines.

Useful defining properties include:

• Parallelogram: Both pairs of opposite sides are parallel.
• Rectangle: A parallelogram with four right angles.
• Rhombus: A parallelogram with four equal sides.
• Square: Four equal sides and four right angles.
• Kite: Two pairs of adjacent sides are equal.
• Trapezoid: At least one pair of opposite sides is parallel.
  (Some textbooks define it as exactly one pair.)

These categories can overlap. For example, every square is also a rectangle, a rhombus, and a parallelogram.

Worked example

Quadrilateral $ABCD$ has:

• $AB=BC=CD=DA$
• $\angle A=\angle B=\angle C=\angle D=90^\circ$

Classify the quadrilateral.

Step 1: Check the side lengths.

All four sides are equal, so the shape has the side property of a rhombus.

Step 2: Check the angles.

All four angles are right angles, so the shape has the angle property of a rectangle.

Step 3: Use both properties together.

A quadrilateral with four equal sides and four right angles is a square.

Step 4: Include the related classifications.

Because a square has four right angles, it is also a rectangle.
Because a square has four equal sides, it is also a rhombus.
A square also has both pairs of opposite sides parallel, so it is a parallelogram.

Therefore, the best classification is

and it is also a rectangle, rhombus, and parallelogram.