Four-sided shapes
A quadrilateral has four sides, four vertices and four angles. Its interior angles always add to 360° (a diagonal splits it into two triangles: 2 × 180° = 360°).
The quadrilateral family
| Shape | Sides | Angles | Diagonals |
|---|---|---|---|
| Parallelogram | opposite sides parallel and equal | opposite angles equal | bisect each other |
| Rectangle | opposite sides parallel and equal | all 90° | equal length, bisect each other |
| Rhombus | all 4 equal, opposite sides parallel | opposite angles equal | perpendicular, bisect each other |
| Square | all 4 equal, opposite sides parallel | all 90° | equal, perpendicular, bisect each other |
| Trapezoid | at least 1 pair of parallel sides | — | — |
| Kite | 2 pairs of adjacent equal sides | one pair of opposite angles equal | perpendicular |
How they’re related
Shapes inherit properties:
- A square is a rectangle and a rhombus and a parallelogram.
- A rectangle is a parallelogram with right angles.
- A rhombus is a parallelogram with equal sides.
So "Is a square a rectangle?" Yes! It has all the properties of a rectangle.
(Some textbooks define a trapezoid as having exactly one pair of parallel sides. Under the "at least one" definition, parallelograms count as trapezoids too.)
Angle facts
- All quadrilaterals: angles add to 360°.
- Parallelograms: consecutive angles add to 180°.
Example: A parallelogram has one angle of 70°. The others are 110°, 70°, 110°.
Example: A quadrilateral has angles 85°, 95° and 110°. The fourth is 360 − 290 = 70°.
Identifying a quadrilateral
Ask:
- How many pairs of parallel sides? (0, 1 or 2)
- Are the sides equal?
- Are there right angles?
- What do the diagonals do?
Worked examples
Example 1: A shape has 4 equal sides and no right angles. Name it.
Rhombus.
Example 2: A kite has angles 120°, 70° and 70°. Find the fourth.
360 − 260 = 100°. (The two 70° angles are the equal pair, where the unequal sides meet.)
Example 3: True or false: every rectangle is a square.
False. A rectangle’s sides don’t have to be equal.
Common mistakes
- Thinking a square isn’t a rectangle.
- Forgetting the angle sum is 360°, not 180°.
- Mixing up rhombus and kite: a rhombus has all sides equal; a kite has two pairs of adjacent equal sides.
Try it yourself
- Angles of a quadrilateral: 90°, 90°, 120°. Fourth?
- Which quadrilaterals always have perpendicular diagonals?
- A parallelogram has one angle of 135°. Find the others.
- Name a quadrilateral with exactly one pair of parallel sides.
Answers: 1) 60° 2) rhombus, square and kite 3) 45°, 135°, 45° 4) trapezoid
Quick recap
- A quadrilateral‘s angles add to 360°.
- Parallelogram, rectangle, rhombus and square have two pairs of parallel sides.
- Trapezoids and kites have their own special properties.