Quadrilaterals and Other Polygons
Theory
The four angles of any quadrilateral add to 360°, and a polygon with \(n\) sides has an angle sum of \((n - 2) \times 180^\circ\). This Year 9 maths page revises the Year 8 properties of quadrilaterals: parallelograms, rhombuses, rectangles, squares, kites and trapeziums, their diagonals, angles with pronumerals, and the interior and exterior angles of regular polygons.
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 polygon is a closed shape with straight sides. A regular polygon has all sides equal and all angles equal. An exterior angle is made by extending one side; at each vertex the interior and exterior angles add to \(180^\circ\). The exterior angles of any convex polygon add to \(360^\circ\).
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:
In a kite, the two angles between the unequal sides are equal.
For a polygon with \(n\) sides, and a regular polygon with \(n\) 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 \(360^\circ\).
Polygons
- Angle sum: count the sides, then use \((n - 2) \times 180^\circ\).
- Missing angle: subtract the known angles from the angle sum. If some angles are equal or written with \(x\), add the expressions, set them equal to the angle sum and solve.
- Regular polygon: each exterior angle is \(360^\circ \div n\); each interior angle is \(180^\circ\) minus that.
- Sides from an angle: get the exterior angle, then \(n = 360^\circ \div \text{exterior angle}\). The answer must be a whole number.
(i) Three angles of a quadrilateral are \(78^\circ\), \(112^\circ\) and \(96^\circ\). Find the fourth angle.
(ii) The angles of a quadrilateral are \(x\), \(x + 20^\circ\), \(2x + 10^\circ\) and \(2x - 30^\circ\). Find \(x\) and each angle.
(i) 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 \(180^\circ\)).
(ii) Add the four expressions and set the total equal to \(360^\circ\).
\(x + (x + 20) + (2x + 10)\) \(+\ (2x - 30) = 360\)
| \(6x\) | \(=\) | \(360\) |
| \(x\) | \(=\) | \(\mathbf{60}\) |
The angles are \(60^\circ\), \(80^\circ\), \(130^\circ\) and \(90^\circ\). Check: \(60 + 80 + 130 + 90 = 360\) \(\checkmark\)
(i) One angle of a parallelogram is \(118^\circ\). Find the other three angles.
(ii) \(PQRS\) is a kite with \(PQ = PS\) and \(RQ = RS\). \(\angle QPS = 58^\circ\) and \(\angle QRS = 84^\circ\). Find the size of \(\angle PQR\).
(i)
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: \(118 + 62 + 118 + 62 = 360\).
(ii)
A kite has one pair of equal angles: the two between the unequal sides, at \(Q\) and \(S\). Call each one \(x\) and use the angle sum of a quadrilateral.
| \(58 + 84 + x + x\) | \(=\) | \(360\) |
| \(142 + 2x\) | \(=\) | \(360\) |
| \(2x\) | \(=\) | \(218\) |
| \(x\) | \(=\) | \(109\) |
\(\therefore\ \angle PQR = \mathbf{109^\circ}\) (angle sum of a quadrilateral, equal angles of a kite).
The \(58^\circ\) and \(84^\circ\) angles sit on the axis of symmetry \(PR\). They are not equal to each other.
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:
| \(\angle ABC\) | \(=\) | \(2 \times 66^\circ\) |
| \(\) | \(=\) | \(\mathbf{132^\circ}\) |
(i) Find the angle sum of a hexagon.
(ii) Find each exterior angle and each interior angle of a regular decagon (\(10\) sides).
(iii) Each exterior angle of a regular polygon is \(24^\circ\). How many sides does it have?
(i) A hexagon has \(6\) sides, so it splits into \(4\) triangles.
| \(S\) | \(=\) | \((6 - 2) \times 180^\circ\) |
| \(\) | \(=\) | \(\mathbf{720^\circ}\) |
(ii) The exterior angles of a regular polygon are equal and add to \(360^\circ\).
| \(\text{exterior}\) | \(=\) | \(\dfrac{360^\circ}{10} = \mathbf{36^\circ}\) |
| \(\text{interior}\) | \(=\) | \(180^\circ - 36^\circ = \mathbf{144^\circ}\) |
Check: \((10 - 2) \times 180^\circ = 1440^\circ\) and \(1440^\circ \div 10 = 144^\circ\) \(\checkmark\)
(iii) Divide \(360^\circ\) by one exterior angle.
| \(n\) | \(=\) | \(\dfrac{360^\circ}{24^\circ} = \mathbf{15}\) |
Each interior angle is \(180^\circ - 24^\circ = 156^\circ\).
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.
How do you find the angle sum of a polygon?
Use \((n-2) \times 180^\circ\), where \(n\) is the number of sides. A pentagon has \(3 \times 180^\circ = 540^\circ\).
How do you find the number of sides of a regular polygon from an angle?
Find the exterior angle (\(180^\circ\) minus the interior angle), then divide \(360^\circ\) by it. An exterior angle of \(24^\circ\) gives \(15\) sides.