Properties of Quadrilaterals
Theory
A property of a special quadrilateral is a reason. Naming it lets you write an equation: in a rectangle the diagonals are equal and bisect each other, so if \(KM = 34\) cm, then \(LN = 34\) cm and each half-diagonal is \(17\) cm. This Year 8 maths page shows how.
Kite: one axis of symmetry; the diagonals meet at right angles, and the axis cuts the other diagonal in half.
Parallelogram: no axis of symmetry; opposite angles are equal; the diagonals bisect each other, cutting it into four triangles.
Rectangle: two axes of symmetry; all the parallelogram properties; the diagonals are equal, so all four half-diagonals are equal and each triangle they make is isosceles.
Rhombus: two axes of symmetry; all the parallelogram properties; the diagonals meet at right angles and bisect the angles they pass through.
Square: four axes of symmetry; all the rhombus properties; the diagonals are equal. A diagonal splits each right angle into two \(45^\circ\) angles.
Two equal things make an equation, with pronumerals on either side. Name the triangle you are working in before using its angle sum.
Hedge the names. A square is both a rectangle and a rhombus, and every rhombus is a parallelogram, but a rhombus is not always a square, and a kite is not always a parallelogram.
Reasons to write (for ABCD with diagonals meeting at O):
| Reason | What it lets you write |
|---|---|
| the diagonals of a parallelogram bisect each other | \(AO = OC\) and \(BO = OD\) |
| the diagonals of a rectangle are equal | \(AC = BD\) |
| the diagonals of a rhombus meet at right angles | \(\angle AOB = 90^\circ\) |
| the diagonals of a rhombus bisect its angles | \(\angle DAC = \angle CAB\) |
| opposite angles of a parallelogram | \(\angle DAB = \angle BCD\) |
| co-interior angles, \(AB \parallel DC\) | \(\angle DAB + \angle ADC = 180^\circ\) |
Axes of symmetry:
| Shape | Axes of symmetry |
|---|---|
| kite | 1 |
| parallelogram | 0 |
| rectangle | 2 |
| rhombus | 2 |
| square | 4 |
How to use a property
- Name the shape and the property you need.
- Write the equation it gives, with the property as the reason.
- Solve, working in one named triangle at a time.
(i) \(LN\)
(ii) \(VL\)
| \(LN\) | \(=\) | \(KM = 34\text{ cm}\) |
| \(VL\) | \(=\) | \(34 \div 2 = 17\text{ cm}\) |
(i) \(34\) cm (the diagonals of a rectangle are equal)
(ii) \(17\) cm (the diagonals bisect each other)
(i) \(\angle XYZ\)
(ii) \(\angle WXV\)
(iii) \(\angle XWV\)
| \(\angle XYZ\) | \(=\) | \(180 - 76 = 104^\circ\) |
| \(\angle WXV\) | \(=\) | \(76 \div 2 = 38^\circ\) |
| \(\angle XWV\) | \(=\) | \(180 - 38 - 90 = 52^\circ\) |
(i) \(104^\circ\) (co-interior angles, \(WX \parallel ZY\))
(ii) \(38^\circ\) (the diagonals of a rhombus bisect its angles)
(iii) \(52^\circ\): \(\angle WVX = 90^\circ\) (the diagonals of a rhombus meet at right angles), then the angle sum of \(\triangle WXV\).
(i) \(p\)
(ii) the perimeter
| \(3p + 4\) | \(=\) | \(5p - 6\) |
| \(10\) | \(=\) | \(2p\) |
| \(p\) | \(=\) | \(5\) |
| \(2 \times 19 + 2 \times 9\) | \(=\) | \(56\) |
(i) \(p = 5\), so \(AB = AD = 19\) cm.
(ii) The perimeter is \(56\) cm.
| \(\angle QPR\) | \(=\) | \(90 \div 2 = 45^\circ\) |
| \(\angle PRQ\) | \(=\) | \(180 - 90 - 45 = 45^\circ\) |
Both are \(45^\circ\): the diagonal bisects the right angle at \(P\), and \(\triangle PQR\) is isosceles and right-angled.
(i) Name it as exactly as you can.
(ii) Is it a rhombus?
(iii) Is it a trapezium?
(i) Four equal sides make a rhombus, so \(EF \parallel HG\) and \(\angle EHG = 180 - 90 = 90^\circ\) (co-interior angles). The opposite angles are also \(90^\circ\), so it is a square.
(ii) Yes: a square has every rhombus property.
(iii) Yes: two pairs of parallel sides is at least one pair.
Common pitfalls
Frequently asked questions
Do the diagonals of a kite bisect each other?
Only one is cut in half: the diagonal along the axis of symmetry cuts the other diagonal into two equal parts.
Are the diagonals of a rhombus equal?
Not usually. They bisect each other at right angles; they are equal only in a square.
Which quadrilaterals have diagonals that meet at right angles?
Kites, rhombuses and squares.
How many axes of symmetry does a rhombus have?
Two: each diagonal is an axis of symmetry.
Is every rhombus a parallelogram?
Yes. Its opposite sides are parallel, so it has every parallelogram property.