Angles

Before you go deeper, let us gather the rules you already met. Every harder problem this year is just these facts used cleverly together.

• Angles on a straight line add up to $180\degree$.
• Angles around a point add up to $360\degree$.
• Vertically opposite angles (where two lines cross) are equal.
• The three angles inside any triangle add up to $180\degree$.

The new skill this year is giving a reason for every step. When you write an angle, also write why — for example, "angles on a line" or "vertically opposite". This habit keeps long problems organised and lets you spot your own mistakes.

So in the diagram, the $110\degree$ angle has an equal angle directly opposite, and each neighbour is $180\degree - 110\degree = 70\degree$ — two facts in one picture.

On parallel lines crossed by a transversal, three pairs of angles are special:

• Corresponding angles (the F shape) are equal.
• Alternate angles (the Z shape) are equal.
• Co-interior angles (the C shape) add up to $180\degree$.

This year, problems rarely use just one rule. You hunt for an F, Z or C shape, find one angle, then use the straight-line or vertically-opposite fact to step across to the next.

In the diagram the top angle is $72\degree$. Its alternate angle on the lower line is also $72\degree$. The angle next to that, between the parallel lines, is co-interior with the top one, so it is $180\degree - 72\degree = 108\degree$.

The trick with messy diagrams is to mark every angle you can, one rule at a time, until the one you want appears. Always note the reason.

Take a triangle and extend one side past a corner. The angle outside, between the extended side and the next side, is an exterior angle.

Here is the handy property: the exterior angle equals the sum of the two opposite interior angles — the two corners it does not touch.

So in the diagram, the exterior angle is $65\degree + 50\degree = 115\degree$.

Why does this work? The exterior angle and the interior angle next to it sit on a straight line, so they add to $180\degree$. The three interior angles also add to $180\degree$. Comparing those two facts shows the exterior angle must equal the other two interior angles.

This shortcut saves a step: instead of finding the third interior angle first, you can jump straight to the exterior angle.

A quadrilateral is any four-sided shape. The four angles inside it always add up to $360\degree$ — and you can see exactly why.

Draw one diagonal across the quadrilateral. It splits the shape into two triangles. Each triangle's angles add to $180\degree$, so together the quadrilateral's angles add to $2 \times 180\degree = 360\degree$.

To find a missing angle, add the three you know and subtract from $360\degree$. If a quadrilateral has angles $85\degree$, $100\degree$, $74\degree$ and $x$, then: $x = 360\degree - 85\degree - 100\degree - 74\degree = 101\degree.$

Special quadrilaterals give extra clues. A parallelogram has two pairs of equal angles; a rectangle has four $90\degree$ angles; a kite has one pair of equal angles. Use those facts to cut down the unknowns.

A polygon is a closed shape with straight sides. The triangle idea keeps working: 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 $(5 - 2) \times 180\degree = 540\degree$. A hexagon ($n = 6$) gives $720\degree$.

A regular polygon has all sides and all angles equal. To find one interior angle, divide the total by the number of angles: $\text{one interior angle} = \frac{(n - 2) \times 180\degree}{n}.$

So each angle of a regular hexagon is $720\degree \div 6 = 120\degree$. For an irregular polygon you still know the total — just subtract the angles you are given.

Each exterior angle of a polygon is the angle you turn through at a corner if you walk all the way around the outside.

Here is the elegant fact: for any polygon, the exterior angles add up to $360\degree$. It does not matter how many sides it has — when you walk the full loop, you complete one whole turn.

For a regular polygon all the exterior angles are equal, so each one is: $\text{one exterior angle} = \frac{360\degree}{n}.$

A regular pentagon has $360\degree \div 5 = 72\degree$ at each corner.

The interior and exterior angle at any corner sit on a straight line, so they add to $180\degree$: $\text{interior angle} = 180\degree - \text{exterior angle}.$

This often gives the quickest route. To find an interior angle of a regular polygon, work out the exterior angle first ($360\degree \div n$), then subtract from $180\degree$.

Getting confident with these angle skills sets you up for a lot of geometry ahead:

• Bearings and constructions — accurate angle reasoning underpins navigation and scale drawings.
• Tessellations — knowing interior angles explains exactly which shapes tile a plane with no gaps.
• Circle theorems — later courses build angle rules involving circles, all resting on triangle and line facts.
• In everyday life, angle reasoning appears in design, architecture, engineering and computer graphics.

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