Classifying Quadrilaterals
Classifying a quadrilateral means collecting its properties, its sides, angles and diagonals, and giving the most exact name they allow. This NSW Year 8 maths subtopic (NESA outcome MA4‑GEO‑C‑01) works with the family of special quadrilaterals: the trapezium, kite, parallelogram, rhombus, rectangle and square.
You will read tick and arrow marks, decide which shapes always have two pairs of parallel sides, tell convex from non-convex shapes, and use co-interior angles adding to \(180^\circ\) to show that sides are parallel. You will also test sides on the number plane with Pythagoras' theorem, find the missing corner of a parallelogram, and decide whether a window frame with unequal diagonals is a true rectangle. It is part of Year 8 Geometry.
Theory
A quadrilateral has four sides and four angles. To classify one, collect its properties (sides, angles, diagonals) and give the most exact name they allow: a square is also a rectangle, a rhombus, a parallelogram, a trapezium and a kite. This Year 8 maths page shows how.
Marks. Tick marks show equal sides, arrow marks show parallel sides, and a small square shows a right angle. Name a quadrilateral by going around its vertices in order.
Trapezium: at least one pair of parallel sides. Kite: two pairs of equal adjacent sides. Parallelogram: opposite sides parallel and equal.
Rectangle: a parallelogram with all angles \(90^\circ\). Rhombus: a parallelogram with all sides equal. Square: a rhombus with all angles \(90^\circ\) (a rectangle with all sides equal).
Most exact name. Give the name that says the most. A trapezium whose other two sides are equal is an isosceles trapezium.
Angles. The angle sum is \(360^\circ\). Every angle of a convex quadrilateral is less than \(180^\circ\); a non-convex one has one reflex angle, so it looks dented.
Parallel from angles. Two angles along one side that add to \(180^\circ\) are co-interior, so the other two sides are parallel.
On the number plane. Sides with the same grid steps are parallel and equal; square and add the steps (Pythagoras’ theorem) to compare lengths. With pronumerals, opposite sides of a parallelogram are equal: \(5a - 2 = 3a + 8\) gives \(a = 5\) and a side of \(23\) cm.
Properties of the special quadrilaterals:
| Shape | Sides and angles | Diagonals |
|---|---|---|
| trapezium | at least 1 pair of parallel sides | |
| kite | 2 pairs of equal adjacent sides; 1 pair of opposite angles equal | perpendicular; one bisected |
| parallelogram | opposite sides parallel and equal; opposite angles equal | bisect each other |
| rectangle | parallelogram with all angles \(90^\circ\) | equal; bisect each other |
| rhombus | parallelogram with 4 equal sides | bisect each other at right angles |
| square | rhombus with all angles \(90^\circ\) | equal; bisect each other at right angles |
How to name a quadrilateral
- Read the marks, or work out the sides, angles and diagonals.
- Check parallel sides (arrows, matching grid steps or co-interior angles).
- Give the most exact name those facts allow, and no more.
Two pairs of equal adjacent sides make \(KLMN\) a kite. No sides are given as parallel, so it is not shown to be a parallelogram.
| \(\angle A + \angle B\) | \(=\) | \(112 + 68 = 180\) |
| \(\angle B + \angle C\) | \(=\) | \(68 + 112 = 180\) |
\(AD \parallel BC\) and \(AB \parallel DC\) (co-interior angles add to \(180^\circ\)). No angle is \(90^\circ\) and nothing is said about the sides, so \(ABCD\) is a parallelogram.
\(PQ\) and \(SR\) are both \(5\) right, \(1\) up; \(PS\) and \(QR\) are both \(1\) right, \(4\) up. Matching steps: a parallelogram.
| \(PR^2\) | \(=\) | \(6^2 + 5^2 = 61\) |
| \(QS^2\) | \(=\) | \(4^2 + 3^2 = 25\) |
| \(PQ^2\) | \(=\) | \(26\) |
| \(QR^2\) | \(=\) | \(17\) |
The diagonals are not equal, so it is not a rectangle, and \(PQ \ne QR\), so it is not a rhombus. \(PQRS\) is a parallelogram.
(i) \(x\)
(ii) the four angles
(iii) whether \(KLMN\) is convex
The four angles add to \(360^\circ\) (angle sum of a quadrilateral). Collect like terms: \(x + x + x + 5x = 8x\) and \(2 + 10 + 28 = 40\).
| \(8x + 40\) | \(=\) | \(360\) |
| \(8x\) | \(=\) | \(320\) |
| \(x\) | \(=\) | \(40\) |
(i) \(x = 40\) (angle sum of a quadrilateral)
(ii) \(40^\circ\), \(42^\circ\), \(50^\circ\) and \(228^\circ\)
(iii) \(228^\circ\) is reflex, so \(KLMN\) is non-convex.
(i) A square is a rectangle.
(ii) A rhombus is a square.
(iii) A parallelogram is a trapezium.
(i) Always: a square is a parallelogram with four right angles.
(ii) Only sometimes: a rhombus needs a right angle to be a square.
(iii) Always: two pairs of parallel sides include at least one pair.
Common pitfalls
Frequently asked questions
What is a quadrilateral?
A closed shape with four straight sides and four angles.
What is the difference between a rhombus and a square?
Both have four equal sides; a square also has four right angles.
Is a square a rectangle?
Yes. A square has four right angles and opposite sides parallel and equal, so it is a special rectangle.
What is a trapezium?
A quadrilateral with at least one pair of parallel sides.
What is a non-convex quadrilateral?
A quadrilateral with one reflex angle (more than \(180^\circ\)), so it looks dented.