Review of Geometry
Theory
This Year 10 maths page revises the angle facts from Years 7 to 9 that every Year 10 geometry proof is built on: corresponding, alternate and co-interior angles on parallel lines, the exterior angle of a triangle, isosceles triangles, the angles of parallelograms and trapeziums, and the angle sum of a polygon \((n-2) \times 180^\circ\).
When a line called a transversal cuts two parallel lines, it makes three special pairs of angles. Corresponding angles (an F shape) are equal. Alternate angles (a Z shape) are equal. Co-interior angles (a C shape, both between the lines on the same side) add to \(180^\circ\).
Some facts hold for any lines: angles on a straight line add to \(180^\circ\), angles at a point add to \(360^\circ\), and vertically opposite angles are equal.
The angles of a triangle add to \(180^\circ\). The exterior angle of a triangle (made by extending one side) equals the sum of the two interior opposite angles. In an isosceles triangle the base angles, opposite the equal sides, are equal. The angles of a quadrilateral add to \(360^\circ\).
In a parallelogram both pairs of opposite sides are parallel, so neighbouring angles are co-interior and add to \(180^\circ\), and opposite angles are equal. A trapezium has one pair of parallel sides, so the two angles along each slanting side are co-interior and add to \(180^\circ\). In an isosceles trapezium the two base angles on each parallel side are also equal. An angle bisector cuts an angle into two equal halves.
A polygon with \(n\) sides has an angle sum of \((n-2) \times 180^\circ\). A regular polygon has all sides equal and all angles equal. Its exterior angles always add to \(360^\circ\), and each interior angle and its exterior angle add to \(180^\circ\).
Always give a reason with each step, such as "alternate angles, \(AB \parallel CD\)".
Angle sum of a polygon with \(n\) sides:
For a regular polygon with \(n\) sides:
| Fact | Rule | Reason to write |
|---|---|---|
| Straight line | add to \(180^\circ\) | angles on a straight line |
| At a point | add to \(360^\circ\) | angles at a point |
| X crossing | equal | vertically opposite angles |
| F shape | equal | corresponding angles, parallel lines |
| Z shape | equal | alternate angles, parallel lines |
| C shape | add to \(180^\circ\) | co-interior angles, parallel lines |
| Triangle | add to \(180^\circ\) | angle sum of a triangle |
| Exterior angle | = sum of the two interior opposite angles | exterior angle of a triangle |
| Isosceles | base angles equal | base angles of an isosceles triangle |
| Quadrilateral | add to \(360^\circ\) | angle sum of a quadrilateral |
| Parallelogram | opposite angles equal; neighbouring angles add to \(180^\circ\) | opposite angles of a parallelogram |
| Trapezium | angles along a slanting side add to \(180^\circ\) | co-interior angles, parallel sides |
How to find an unknown angle
- Mark every angle you know on the diagram, including equal sides and parallel-line arrows.
- Look for a shape: an F, Z or C on parallel lines, a straight line, a point, a triangle or a polygon.
- Find one angle at a time, writing the reason next to it.
- Form an equation if the angles are written with a pronumeral, then solve it.
- Check the angles fit: a straight line still adds to \(180^\circ\), a triangle to \(180^\circ\).
Useful constructions and tricks
- A bend between parallel lines: draw a third parallel line through the corner and split the angle in two (Example 4).
- Two shapes meeting at a point: find each interior angle, then use angles at a point. A regular pentagon (\(108^\circ\)) and a square (\(90^\circ\)) leave \(360^\circ - 108^\circ - 90^\circ = 162^\circ\).
- A diagonal of a regular polygon: two equal sides make an isosceles triangle, so the two base angles are equal.
- Parallelogram or trapezium angles with \(x\): set opposite angles of a parallelogram equal, or make co-interior angles add to \(180^\circ\).
- An angle bisector: find the whole angle first (often from the angle sum of a triangle), halve it, then use the angle sum of the smaller triangle.
- Polygon angles with \(x\): add all the angle expressions and set them equal to \((n-2) \times 180\).
- Interior angle is \(k\) times the exterior: write \(e + ke = 180^\circ\), find \(e\), then \(n = \dfrac{360^\circ}{e}\).
\(a\) sits in the same position at the other intersection: an F shape.
| \(a\) | \(=\) | \(\mathbf{63^\circ}\ \text{(corresponding, parallel lines)}\) |
\(a\) and \(b\) make a straight line.
| \(b\) | \(=\) | \(180^\circ - 63^\circ\) |
| \(\) | \(=\) | \(\mathbf{117^\circ}\ \text{(straight angle)}\) |
\(c\) and the \(63^\circ\) angle are opposite each other at a crossing.
| \(c\) | \(=\) | \(\mathbf{63^\circ}\ \text{(vertically opposite)}\) |
Check: \(b\) and \(c\) are co-interior and \(117^\circ + 63^\circ = 180^\circ\).
Both angles are between the parallel lines, on opposite sides of the transversal: a Z shape. Alternate angles are equal, so make an equation.
| \(5x - 22\) | \(=\) | \(3x + 18\) |
| \(2x\) | \(=\) | \(40\) |
| \(x\) | \(=\) | \(\mathbf{20}\) |
Each angle is \(5 \times 20 - 22 = \mathbf{78^\circ}\). Check: \(3 \times 20 + 18 = 78\).
If the angles were co-interior instead, the equation would be \((5x - 22) + (3x + 18) = 180\).
(i) Find the angle sum of a nonagon (\(9\) sides).
(ii) Find each interior and exterior angle of a regular nonagon.
(iii) Each interior angle of another regular polygon is \(156^\circ\). How many sides does it have?
(i)
| \(S\) | \(=\) | \((9 - 2) \times 180^\circ\) |
| \(\) | \(=\) | \(\mathbf{1260^\circ}\) |
(ii) The angles are equal, so share the sum. The exterior angles add to \(360^\circ\).
| \(\text{interior}\) | \(=\) | \(\dfrac{1260^\circ}{9} = \mathbf{140^\circ}\) |
| \(\text{exterior}\) | \(=\) | \(\dfrac{360^\circ}{9} = \mathbf{40^\circ}\) |
(iii) Work through the exterior angle.
| \(\text{exterior}\) | \(=\) | \(180^\circ - 156^\circ = 24^\circ\) |
| \(n\) | \(=\) | \(\dfrac{360^\circ}{24^\circ} = \mathbf{15}\) |
No single transversal joins the two given angles. Draw a third line \(EF\) through \(E\), parallel to \(AB\) and \(CD\). It splits \(x\) into two parts.
| \(\angle BEF\) | \(=\) | \(36^\circ\ \text{(alternate, } AB \parallel EF)\) |
| \(\angle DEF\) | \(=\) | \(41^\circ\ \text{(alternate, } CD \parallel EF)\) |
| \(x\) | \(=\) | \(36^\circ + 41^\circ = \mathbf{77^\circ}\) |
If the given angles open the other way (co-interior with \(EF\)), find each part as \(180^\circ\) minus the given angle, then add.
Common pitfalls
Frequently asked questions
What are alternate angles?
Alternate angles are between two parallel lines on opposite sides of the transversal, in a Z shape. They are equal.
What is the difference between co-interior and corresponding angles?
Corresponding angles are in matching positions at each intersection (an F shape) and are equal. Co-interior angles are both between the parallel lines on the same side (a C shape) and add to \(180^\circ\).
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.
What is the exterior angle of a triangle equal to?
It equals the sum of the two interior opposite angles. If the two angles not next to it are \(35^\circ\) and \(70^\circ\), the exterior angle is \(105^\circ\).