Angles

By Stage 9 a single angle rule is rarely enough. A good problem hands you one angle and asks you to reach another several steps away. The skill is chaining rules together and writing a reason beside every angle.

The toolkit you carry into every problem:

• Angles on a straight line add up to $180\degree$; angles around a point add up to $360\degree$.
• Vertically opposite angles are equal.
• On parallel lines: corresponding angles (F shape) are equal, alternate angles (Z shape) are equal, co-interior angles (C shape) add to $180\degree$.
• The three angles of any triangle add up to $180\degree$.

In the diagram the $68\degree$ angle on the top line has an alternate angle of $68\degree$ inside the triangle. With a known angle of $48\degree$ at the other corner, the triangle rule finishes the job: $180\degree - 68\degree - 48\degree = 64\degree$. Two rules, one tidy answer — and a reason written at each step.

A polygon is a closed shape with straight sides. To find the total of its interior angles, pick one corner and draw diagonals to every other corner. A polygon with $n$ sides splits into $(n - 2)$ triangles.

So the interior angle sum of a polygon with $n$ sides is: $S = (n - 2) \times 180\degree.$

A pentagon ($n = 5$) gives $540\degree$; a hexagon ($n = 6$) gives $720\degree$; an octagon ($n = 8$) gives $1080\degree$.

A regular polygon has all sides and all angles equal, so one interior angle is the total shared out: $\text{one interior angle} = \frac{(n - 2) \times 180\degree}{n}.$

Each angle of a regular hexagon is $720\degree \div 6 = 120\degree$. For an irregular polygon you still know the total $S$ — simply subtract the angles you are given to find a missing one.

Each exterior angle of a polygon is the angle you turn through at a corner when you walk around the outside. Walk the whole loop and you complete exactly one full turn, so: $\text{sum of exterior angles} = 360\degree.$

This holds for every polygon, no matter how many sides. For a regular polygon every exterior angle is the same, so: $\text{one exterior angle} = \frac{360\degree}{n}.$

A regular octagon has $360\degree \div 8 = 45\degree$ at each corner. The interior and exterior angle at a corner sit on a straight line, so $\text{interior} = 180\degree - \text{exterior}$.

The real Stage 9 power is working backwards. If you know a regular polygon has an exterior angle of $24\degree$, then $n = 360\degree \div 24\degree = 15$ sides. If you are given an interior angle of $150\degree$, first find the exterior angle, $180\degree - 150\degree = 30\degree$, then $n = 360\degree \div 30\degree = 12$ sides.

Pythagoras' theorem is one of the most famous results in mathematics, and it works for every right-angled triangle. It connects the three side lengths.

The longest side, always opposite the right angle, is the hypotenuse. Call it $c$, and the two shorter sides $a$ and $b$. Then: $a^2 + b^2 = c^2.$

To find the hypotenuse, add the squares of the two shorter sides and take the square root: $c = \sqrt{a^2 + b^2}.$

To find a shorter side, rearrange — subtract instead of add: $a = \sqrt{c^2 - b^2}.$

For a triangle with shorter sides $6\text{ cm}$ and $8\text{ cm}$: $c = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10\text{ cm}$.

A quick check: the hypotenuse must be the longest side. If your answer for $c$ comes out shorter than one of the other sides, something has gone wrong.

Once you trust Pythagoras' theorem, the skill becomes spotting the right-angled triangle inside a problem. Ladders, ramps, screen sizes, the diagonal of a rectangle — they all hide a right angle.

The reliable method:

1. Sketch the situation and mark the right angle.
2. Label the hypotenuse $c$ and the two shorter sides $a$ and $b$.
3. Decide whether you are finding the hypotenuse (add the squares) or a shorter side (subtract).
4. Take the square root and round sensibly.

A $5\text{ m}$ ladder rests against a wall with its foot $1.5\text{ m}$ from the base. The ladder is the hypotenuse, and the height $h$ is a shorter side, so subtract: $h = \sqrt{5^2 - 1.5^2} = \sqrt{25 - 2.25} = \sqrt{22.75} \approx 4.77\text{ m}.$

The diagonal of a rectangle works the same way: the two sides are $a$ and $b$, and the diagonal is the hypotenuse $c$.

Getting confident with these angle skills and Pythagoras' theorem sets you up well for the geometry ahead:

• Trigonometry — the next step after Pythagoras, using angles to find sides in right-angled triangles.
• Circle theorems — later courses build powerful angle rules inside circles, all resting on triangle and line facts.
• Coordinate geometry — Pythagoras gives the distance between two points on a grid.
• In everyday life, this reasoning appears in construction, navigation, design, engineering and computer graphics.

If a later geometry topic feels tricky, there is usually an angle fact or Pythagoras hiding underneath that needs a quick refresh. Keep this guide handy — multi-step problems get faster every time you practise.