Square
- four equal sides;
- four right angles;
- two pairs of parallel sides;
- equal diagonals that bisect at 90°.
Learning Objectives
- define a quadrilateral;
- identify common types of quadrilaterals;
- state their side, angle and diagonal properties;
- use the 360° angle sum to solve problems;
- compare quadrilaterals accurately.
Introduction
This lesson develops Quadrilaterals from meaning and patterns before moving to calculations. Look for what the quantities, shapes or symbols represent, then connect that idea to the method.
Key Notes
Read the definitions and key facts below carefully. Each rule is most useful when you can explain why it fits the situation, not simply recall it.
🌍 Why This Matters
Quadrilaterals appear in windows, doors, books, screens, tiles, roads, roofs and building plans. Architects, engineers and designers use their properties to create stable and accurate structures.
What Is a Quadrilateral?
A quadrilateral is a polygon with four sides, four vertices and four interior angles.
The interior angles of every quadrilateral add up to 360°.
Six Important Quadrilaterals
Rectangle
- opposite sides equal and parallel;
- four right angles;
- equal diagonals;
- diagonals bisect each other.
Parallelogram
- opposite sides equal and parallel;
- opposite angles equal;
- adjacent angles add to 180°;
- diagonals bisect each other.
Rhombus
- four equal sides;
- opposite angles equal;
- opposite sides parallel;
- diagonals meet at 90°.
Kite
- two pairs of equal adjacent sides;
- one pair of opposite angles equal;
- diagonals meet at 90°;
- one diagonal bisects the other.
Trapezium
- one pair of parallel sides;
- an isosceles trapezium has equal non-parallel sides;
- its base angles are equal.
Quick Comparison Table
| Shape | Equal sides | Right angles | Parallel sides | Equal diagonals |
|---|---|---|---|---|
| Square | Four | Four | Two pairs | Yes |
| Rectangle | Opposite | Four | Two pairs | Yes |
| Parallelogram | Opposite | Not always | Two pairs | Not always |
| Rhombus | Four | Not always | Two pairs | Not always |
| Kite | Adjacent pairs | Not always | Usually none | No |
| Trapezium | Not generally | Not always | One pair | Not generally |
Worked Examples
Example 1
A quadrilateral has angles 82°, 91°, 105° and x.
82+91+105+x=360, so x=82°.
Example 2
The angles are x, 2x, 95° and 85°.
x+2x+95+85=360, so x=60°.
Example 3
One angle of a parallelogram is 68°.
The other angles are 112°, 68° and 112°.
Common Mistakes
A square is a special rectangle because it has four right angles.
A square is also a special rhombus because all four sides are equal.
Try It: Missing Angle in a Quadrilateral
🧠 Let’s Think Together
Before calculating, identify the information given, the result required and the mathematical relationship that connects them. Predict whether the answer should be larger, smaller or unchanged, then use that prediction to check the result.
Practice Questions
Foundation- Define a quadrilateral.
- State the sum of its interior angles.
- Which quadrilateral has four equal sides and four right angles?
- Which quadrilateral has one pair of parallel sides?
- Find x if the angles are 75°, 94°, 118° and x.
- One angle of a parallelogram is 48°. Find the other three.
- The angles are 2x, x, 90° and 120°. Find x.
- The angles are 3x, 2x+10, x−20 and 80°. Find x and all four angles.
- Explain why every square is a rectangle but not every rectangle is a square.
✅ Check Your Work
Show answers and working
- A polygon with four sides.
- 360°.
- Square.
- Trapezium.
- 73°.
- 48°, 132°, 48°, 132°.
- x=50°.
- x=48⅓°. Angles: 145°, 106⅔°, 28⅓° and 80°.
- A square has all rectangle properties, but a rectangle does not require four equal sides.
Exam Focus
Show the mathematical relationship you are using, keep notation consistent and give a clear final answer. Use a quick estimate, diagram or inverse operation to check that the result is reasonable.
Summary
- A quadrilateral has four sides.
- Its interior angles add to 360°.
- Each quadrilateral has distinctive side, angle and diagonal properties.
- Opposite angles in a parallelogram are equal.
- Adjacent angles in a parallelogram add to 180°.