Angles, Polygons & Geometric Reasoning

Angle Facts You Must Know

Geometry questions reward clear reasoning, not just the right number. Every mark in an "angle chase" comes from a correct value and a correct reason. Start from the basic facts and quote them by name.

Angles on a straight line add up to $180°$.
Angles around a point add up to $360°$.
Vertically opposite angles (formed when two straight lines cross) are equal.

Key terms

Vertically opposite — the pair of equal angles directly across an X-shaped crossing.

Transversal — a straight line that cuts across two (usually parallel) lines.

Interior angle — an angle inside a polygon at a vertex.

Exterior angle — the angle between a side and the extension of the next side.

Angles in Parallel Lines

When a transversal crosses parallel lines, three named pairs appear. Parallel lines are marked with matching arrowheads.

Corresponding angles are equal — same position at each crossing (an "F-shape").
Alternate angles are equal — opposite sides of the transversal, between the parallels (a "Z-shape").
Co-interior angles add up to $180°$ — same side of the transversal, between the parallels (a "C-shape"). Also called allied angles.

Watch out

Don't guess which rule applies from the look of the diagram. Trace the F, Z or C shape with your finger first, then name the rule. Co-interior is the only pair that sums to $180°$ — the other two are equal.

Triangles and Quadrilaterals

Angle properties you can quote as reasons:

Angles in a triangle sum to $180°$.
Angles in a quadrilateral sum to $360°$.
An isosceles triangle has two equal sides and two equal base angles.
An equilateral triangle has all angles $60°$.
The exterior angle of a triangle equals the sum of the two opposite interior angles.

Polygons: Interior and Exterior Angles

A polygon has $n$ sides. Splitting it into triangles from one vertex gives $(n-2)$ triangles, so:

$$\text{Sum of interior angles} = (n-2)\times 180°$$

The exterior angles of any polygon always sum to $360°$, no matter how many sides.

At each vertex the interior and exterior angles lie on a straight line, so:

$$\text{interior angle} + \text{exterior angle} = 180°$$

Regular Polygons

A regular polygon has all sides and all angles equal. Because the $n$ exterior angles are equal and sum to $360°$:

$$\text{Exterior angle} = \frac{360°}{n} \qquad \text{Interior angle} = 180° - \frac{360°}{n}$$

Exam tip

For regular polygons, work with the exterior angle first — it is just $360° \div n$. To find $n$ from an interior angle, find the exterior angle ($180° -$ interior) and divide it into $360°$. This is faster and less error-prone than the $(n-2)\times180°$ formula.

Worked example

The interior angle of a regular polygon is $156°$. How many sides does it have?

Exterior angle $= 180° - 156° = 24°$.

$n = \dfrac{360°}{24°} = 15$ sides.

Worked example

Find the sum of the interior angles of a regular octagon, and the size of each interior angle.

Sum $= (n-2)\times 180° = (8-2)\times 180° = 6 \times 180° = 1080°$.

Each interior angle $= \dfrac{1080°}{8} = 135°$.

Check: exterior angle $= 360° \div 8 = 45°$, and $180° - 45° = 135°$. ✓

Bearings

A bearing describes a direction as an angle measured:

1. from North,
2. clockwise,
3. written with three figures (e.g. $072°$, $135°$, $310°$).

Worked example

The bearing of B from A is $063°$. Find the bearing of A from B (the back bearing).

A back bearing differs by $180°$. Since $063° < 180°$, add:

$063° + 180° = 243°$.

So the bearing of A from B is $243°$.

The North lines at A and B are parallel, so the rule is just co-interior / alternate angle reasoning in disguise: if the bearing is under $180°$, add $180°$; if it is $180°$ or more, subtract $180°$.

Writing Geometric Reasons in Proofs

Higher tier "prove that" and "give a reason" questions expect the exact wording. Each step is value + reason.

Exam tip

Never write a reason as "obvious" or "they look the same". Write the named fact in full. A correct angle with a missing or wrong reason usually scores only half the marks. Set work out line by line, quoting one fact per line.

Worked example

In the diagram, $AB$ is parallel to $CD$. A transversal makes an angle of $58°$ with $AB$. Find the co-interior angle $x$ on $CD$, giving a reason.

$x = 180° - 58° = 122°$ (co-interior angles sum to $180°$).

Quick Recap

Line $=180°$, point $=360°$, vertically opposite are equal.
Parallel lines: alternate and corresponding equal; co-interior sum to $180°$.
Polygon interior sum $=(n-2)\times180°$; exterior sum $=360°$ always.
Regular: exterior $=360°/n$, interior $=180°-360°/n$.
Bearings: from North, clockwise, three figures; back bearing $\pm180°$.
Always pair every value with its named reason.