Quadrilateral Geometry
The angles of any quadrilateral add to \(360^\circ\), and each special quadrilateral, from the parallelogram to the kite, has its own properties. This NSW Year 9 Core maths subtopic revises Stage 4 work under NESA outcome MA4‑GEO‑C‑01, with diagonals added to most figures.
You will find missing angles from the angle sum and from angles written as \(x\), \(2x\), \(3x\) and \(4x\), use the equal opposite angles of a parallelogram, and use the properties of diagonals: in a rhombus they cross at right angles and cut the corner angles in half. You will also find angles inside the triangles a diagonal creates using isosceles triangles and parallel lines, deal with a kite's one pair of equal angles, and name a quadrilateral from facts about its diagonals. It is part of Year 9 Core Properties of Geometric Figures.
Theory
The four angles of any quadrilateral add to \(360^\circ\). The special quadrilaterals add their own properties: equal opposite angles in a parallelogram, diagonals at right angles in a rhombus, equal diagonals in a rectangle. This Year 9 maths page revises those properties and uses them to find unknown angles.
A quadrilateral is a shape with four straight sides and four angles. It is named by its vertices in order around the shape, so \(ABCD\) and \(ADCB\) are the same quadrilateral but \(ACBD\) is not.
The angle sum of any quadrilateral is \(360^\circ\). A diagonal joins two opposite vertices and cuts the shape into two triangles of \(180^\circ\) each.
A trapezium has one pair of parallel sides. A parallelogram has two pairs of parallel sides. A rectangle is a parallelogram with four right angles. A rhombus is a parallelogram with four equal sides. A square is both a rectangle and a rhombus. A kite has two pairs of equal sides next to each other.
A shape can have more than one name. A square has every property of a rectangle and of a rhombus, so it is both.
For any quadrilateral with angles \(a\), \(b\), \(c\) and \(d\):
In a parallelogram, adjacent angles are co-interior angles between parallel sides:
| Shape | Its diagonals |
|---|---|
| trapezium | nothing special |
| kite | cross at right angles; one cuts the other in half |
| parallelogram | bisect each other |
| rectangle | bisect each other and are equal |
| rhombus | bisect each other at right angles and bisect the angles |
| square | equal, bisect each other at right angles and bisect the angles |
How to find an unknown angle in a quadrilateral
- Name the shape. Look for marked parallel sides, equal sides and right angles.
- List the properties you can use: the \(360^\circ\) angle sum, equal opposite angles, adjacent angles adding to \(180^\circ\), diagonals at \(90^\circ\), or a diagonal cutting an angle in half.
- Find a triangle. If a diagonal is drawn, work in one triangle at a time with the \(180^\circ\) angle sum.
- Write an equation and solve it. Give a reason for each step.
- Check that the four angles of the quadrilateral add to \(360^\circ\).
The four angles of any quadrilateral add to \(360^\circ\). Add the three you know, then subtract from \(360^\circ\).
| \(78^\circ + 112^\circ + 96^\circ\) | \(=\) | \(286^\circ\) |
| \(x\) | \(=\) | \(360^\circ - 286^\circ\) |
| \(\) | \(=\) | \(\mathbf{74^\circ}\) |
Check: \(78 + 112 + 96 + 74 = 360\). The same rule works when the shape has a dent and one angle is reflex (more than \(180^\circ\)).
The opposite angle is equal, so it is also \(118^\circ\). An adjacent angle is co-interior with it (between the parallel sides), so they add to \(180^\circ\).
| \(118^\circ + x\) | \(=\) | \(180^\circ\) |
| \(x\) | \(=\) | \(\mathbf{62^\circ}\) |
The angle opposite \(x\) is also \(62^\circ\). The angles are \(\mathbf{118^\circ}\), \(\mathbf{62^\circ}\), \(\mathbf{118^\circ}\) and \(\mathbf{62^\circ}\). Check: \(118 + 62 + 118 + 62 = 360\).
The diagonals of a rhombus cross at right angles, so \(\angle AOB = 90^\circ\). Use the angle sum of \(\triangle AOB\).
| \(24^\circ + 90^\circ + \angle ABO\) | \(=\) | \(180^\circ\) |
| \(\angle ABO\) | \(=\) | \(180^\circ - 114^\circ\) |
| \(\) | \(=\) | \(\mathbf{66^\circ}\) |
Diagonal \(BD\) cuts \(\angle ABC\) in half, so double it:
| \(\angle ABC\) | \(=\) | \(2 \times 66^\circ\) |
| \(\) | \(=\) | \(\mathbf{132^\circ}\) |
Take the properties one at a time and see which shapes are left.
| \(\text{they bisect}\) | \(\Rightarrow\) | \(\text{parallelogram}\) |
| \(\text{also equal}\) | \(\Rightarrow\) | \(\text{rectangle or square}\) |
| \(\text{not at } 90^\circ\) | \(\Rightarrow\) | \(\text{not a square}\) |
It is a rectangle. Because the half-diagonals are equal, each triangle at \(O\) is isosceles: its two base angles are equal.
Common pitfalls
Frequently asked questions
What do the angles in a quadrilateral add up to?
\(360^\circ\). A diagonal splits any quadrilateral into two triangles, and each triangle has angles adding to \(180^\circ\).
What are the angle properties of a parallelogram?
Opposite angles are equal, and any two adjacent angles add to \(180^\circ\) because they are co-interior angles between parallel sides.
Do the diagonals of a rhombus meet at right angles?
Yes. The diagonals of a rhombus bisect each other at right angles, and they also cut each corner angle in half.
Is a square a rectangle?
Yes. A square has four right angles and equal diagonals, which are all the properties of a rectangle. It is also a rhombus and a parallelogram.
What is the difference between a rhombus and a rectangle?
A rhombus has four equal sides and diagonals at right angles. A rectangle has four right angles and equal diagonals. A square is both.