Quadrilaterals

The Quadrilateral Family

A quadrilateral is any four-sided polygon, and its interior angles always sum to 360^{\circ} (from the formula (4-2)\times 180^{\circ}). Within this family are several special shapes, each with its own extra properties.

Parallelogram

A parallelogram is a quadrilateral with two pairs of parallel sides. Its key properties:

• Opposite sides are equal in length.
• Opposite angles are equal in measure.
• Consecutive angles (next to each other) are supplementary, summing to 180^{\circ}.
• The diagonals bisect each other (cross at each one's midpoint).

Rectangle

A rectangle is a parallelogram where all four angles are right angles (90^{\circ} each). It keeps all the parallelogram properties above, plus its diagonals are equal in length.

Rhombus

A rhombus is a parallelogram where all four sides are equal in length. It keeps all the parallelogram properties, plus its diagonals are perpendicular to each other and bisect the angles at each vertex.

Square

A square is both a rectangle and a rhombus at once: four equal sides, four right angles, and diagonals that are equal, perpendicular, and bisect each other.

Trapezoid

A trapezoid is a quadrilateral with exactly one pair of parallel sides (called the bases). The non-parallel sides are called legs. If the two legs are equal in length, it's called an isosceles trapezoid, which has the extra property that its base angles are equal in pairs, and its diagonals are equal in length.

How the Shapes Relate

It helps to picture these as nested categories: every square is a rhombus, and every square is a rectangle. Every rectangle and every rhombus is a parallelogram. Every parallelogram is a quadrilateral. But the relationships don't go backward — not every parallelogram is a rectangle, and not every quadrilateral is a trapezoid.

Example

In parallelogram ABCD, angle A measures 72^{\circ}. Find the measures of angles B, C, and D.

1. Angle B is consecutive to angle A, so they're supplementary: B = 180^{\circ} - 72^{\circ} = 108^{\circ}
2. Angle C is opposite angle A, so opposite angles are equal: C = 72^{\circ}
3. Angle D is opposite angle B, so: D = 108^{\circ}
4. Check: 72 + 108 + 72 + 108 = 360^{\circ}, matching the quadrilateral angle sum. ✓

Key Takeaways

• Every quadrilateral's interior angles sum to 360^{\circ}.
• Parallelograms: opposite sides and angles equal, consecutive angles supplementary, diagonals bisect each other.
• Rectangles add equal diagonals and right angles; rhombi add equal sides and perpendicular diagonals; squares combine both.
• Trapezoids have exactly one pair of parallel sides; isosceles trapezoids have equal legs, equal base angles, and equal diagonals.