Properties of Quadrilaterals
Every question with a fully worked solution.
- Properties of Quadrilaterals - Video - Properties of Quadrilaterals Watch
Theory
The properties of a quadrilateral are facts about its sides, angles, diagonals and axes of symmetry. A rectangle, a rhombus and a square are special parallelograms, so they have every property of a parallelogram and more. This Year 7 maths page has the full property table and uses the properties to name shapes and find lengths and angles.
The family shares. A rectangle, a rhombus and a square are special parallelograms, so they have every property of a parallelogram. A square has every property in the table.
Diagonal words. The diagonals bisect each other when each one cuts the other into two equal halves. Equal diagonals have the same length. Diagonals that meet at right angles are perpendicular. A diagonal bisects an angle when it cuts that corner into two equal parts.
Kite. Its axis of symmetry is a diagonal. That diagonal cuts the other diagonal in half at right angles and bisects the two angles it passes through.
Trapezium. It has a pair of parallel sides. (In an isosceles trapezium the non-parallel sides are equal and it has one axis of symmetry.)
Using a property. Write the property as the reason. If the diagonals of rectangle \(ABCD\) meet at \(O\), then \(OB = \dfrac{1}{2} \times BD\) (the diagonals of a rectangle bisect each other).
The property table (a tick means always true):
| Property | Parallelogram | Rectangle |
|---|---|---|
| Opposite sides are parallel | ✓ | ✓ |
| Opposite sides are equal | ✓ | ✓ |
| All four sides are equal | ||
| All angles are right angles | ✓ | |
| Opposite angles are equal | ✓ | ✓ |
| Diagonals bisect each other | ✓ | ✓ |
| Diagonals are equal | ✓ | |
| Diagonals meet at right angles | ||
| Diagonals bisect the angles | ||
| Axes of symmetry | 0 | 2 |
| Property | Rhombus | Square |
|---|---|---|
| Opposite sides are parallel | ✓ | ✓ |
| Opposite sides are equal | ✓ | ✓ |
| All four sides are equal | ✓ | ✓ |
| All angles are right angles | ✓ | |
| Opposite angles are equal | ✓ | ✓ |
| Diagonals bisect each other | ✓ | ✓ |
| Diagonals are equal | ✓ | |
| Diagonals meet at right angles | ✓ | ✓ |
| Diagonals bisect the angles | ✓ | ✓ |
| Axes of symmetry | 2 | 4 |
| Property | Kite | Trapezium |
|---|---|---|
| Opposite sides are parallel | one pair (at least) | |
| Opposite sides are equal | ||
| All four sides are equal | ||
| All angles are right angles | ||
| Opposite angles are equal | one pair | |
| Diagonals bisect each other | ||
| Diagonals are equal | ||
| Diagonals meet at right angles | ✓ | |
| Diagonals bisect the angles | one diagonal | |
| Axes of symmetry | 1 | 0 |
How to name a quadrilateral from its properties
- Find the shapes that have every property listed (use the table).
- Rule out shapes that would need a property that is not given.
- Choose the most exact name left.
How to find a length or angle
- Choose the property that links what you know to what you need.
- Write it as the reason, then work it out.
(i) diagonals that bisect each other but are not equal and do not meet at right angles
(ii) all angles equal and all sides equal
(iii) two pairs of adjacent sides equal and diagonals that meet at right angles?
(i) a parallelogram
(ii) a square
(iii) a kite
(i) all angles equal and opposite sides equal (but not all sides equal)
(ii) all sides equal and opposite angles equal (but not all angles equal)
(iii) opposite sides equal and opposite angles equal (but not all sides or all angles equal)?
(i) a rectangle
(ii) a rhombus
(iii) a parallelogram
(i) \(AC = BD\)
(ii) \(AO = OC\)
(iii) \(AC \perp BD\)
(i) No: equal diagonals are only certain in a rectangle or square.
(ii) Yes: a rhombus is a parallelogram, so its diagonals bisect each other.
(iii) Yes: the diagonals of a rhombus meet at right angles.
(i) \(DE\)
(ii) \(AC\)
(iii) \(CD\)
| \(DE\) | \(=\) | \(BE = 7.5\) |
| \(BD\) | \(=\) | \(7.5 + 7.5 = 15\) |
| \(AC\) | \(=\) | \(BD = 15\) |
(i) \(7.5\) cm (the diagonals bisect each other)
(ii) \(15\) cm (the diagonals of a rectangle are equal)
(iii) \(12\) cm (opposite sides of a rectangle are equal)
(i) \(\angle AOB\)
(ii) \(x\)
(iii) \(\angle BAD\)
| \(x + 90 + 38\) | \(=\) | \(180\) |
| \(x\) | \(=\) | \(52\) |
| \(\angle BAD\) | \(=\) | \(2 \times 38 = 76\) |
(i) \(90^\circ\) (the diagonals of a kite meet at right angles)
(ii) \(x = 52\) (angle sum of \(\triangle AOB\))
(iii) \(76^\circ\) (the axis \(AC\) bisects \(\angle BAD\))
(i) \(x\)
(ii) the perimeter
(iii) \(\angle QPR\)
| \(2x + 3\) | \(=\) | \(17\) |
| \(2x\) | \(=\) | \(14\) |
| \(x\) | \(=\) | \(7\) |
| \(4 \times 17\) | \(=\) | \(68\) |
| \(76 \div 2\) | \(=\) | \(38\) |
(i) \(x = 7\) (all sides of a rhombus are equal)
(ii) \(68\) cm
(iii) \(38^\circ\) (the diagonals of a rhombus bisect its angles)
Common pitfalls
Frequently asked questions
Which quadrilaterals have diagonals that bisect each other?
The parallelogram, rectangle, rhombus and square.
Which quadrilaterals have equal diagonals?
The rectangle and the square (and the isosceles trapezium).
Which quadrilaterals have diagonals at right angles?
The rhombus, the square and the kite.
How many axes of symmetry does a parallelogram have?
None, unless it is also a rectangle or a rhombus (then it has \(2\), or \(4\) for a square).
What property is true for every square but not every rectangle?
All four sides are equal (and the diagonals meet at right angles).