Angles

Before any clever reasoning, you need three small facts that you will use in almost every angle question.

• Angles on a straight line add up to $180\degree$.
• Angles around a point add up to $360\degree$.
• When two straight lines cross, the vertically opposite angles are equal.

These rules feel obvious, but they are the workhorses of angle reasoning. Whenever you see two angles meeting on a line, you can subtract from $180\degree$ to find the missing one. Whenever several angles meet at a single point, subtract from $360\degree$.

A handy detail: at a crossing, every neighbouring pair adds to $180\degree$ (because they sit on a straight line), and every opposite pair is equal. Once you know one of the four angles, you know all four.

When a single straight line (a transversal) cuts two parallel lines, three useful rules pop up:

• Corresponding angles are equal — they make an F shape.
• Alternate angles are equal — they make a Z shape.
• Co-interior angles add to $180\degree$ — they make a C shape.

These rules only work when the lines are truly parallel, so always look for the matching arrow marks before you use them.

A typical question gives you one angle and asks for another several steps away. Pick the matching shape — F, Z or C — and write the rule beside your answer, for example "alternate angles, parallel lines". A short reason turns a guess into proper mathematics.

Two of the most reliable angle facts in your toolbox come from the smallest polygons.

• The three angles of any triangle add up to $180\degree$.
• The four angles of any quadrilateral add up to $360\degree$.

Both follow from a beautiful idea: any quadrilateral splits into two triangles with a single diagonal, so its angles total $2 \times 180\degree = 360\degree$.

When a triangle is isosceles, the two equal sides face two equal angles — a useful shortcut. An equilateral triangle has three sides equal and three $60\degree$ angles. A right-angled triangle has one $90\degree$ angle, so the other two add to $90\degree$.

Special quadrilaterals add their own facts: in a parallelogram the opposite angles are equal and neighbouring angles add to $180\degree$; in a rectangle or square every corner is $90\degree$.

A polygon is a closed shape made from straight sides. To find the total of all 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 any polygon with $n$ sides is: $S = (n - 2) \times 180\degree.$

A triangle gives $180\degree$, a quadrilateral $360\degree$, a pentagon $540\degree$, a hexagon $720\degree$, an octagon $1080\degree$.

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 interior angle of a regular hexagon is $720\degree \div 6 = 120\degree$. For an irregular polygon you still know the total $S$ — 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 as you walk around the outside. Walk the whole way around and you complete exactly one full turn, so for every polygon: $\text{sum of exterior angles} = 360\degree.$

This is true whether the polygon has 5 sides or 50, and whether it is regular or irregular. 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: $\text{interior} = 180\degree - \text{exterior}.$

You can also work backwards. If a regular polygon has an exterior angle of $24\degree$, then $n = 360\degree \div 24\degree = 15$ sides. 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.

By Checkpoint, the most interesting problems give you one angle and ask you to reach another several steps away. The skill is chaining rules together and writing a reason for each one.

A reliable method:

1. Mark every angle you can find from the picture.
2. Add a reason in words: "angles on a straight line", "alternate angles, parallel lines", "isosceles triangle", and so on.
3. Step from a known angle to the next one using the most useful rule.
4. Keep going until you have the angle you were asked for.

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 written reason at every step.

Strong angle skills now will pay you back for years. As soon as you have the basics fluent, you will see them everywhere:

• Pythagoras' theorem and trigonometry rely on knowing the right-angle and triangle facts.
• Circle theorems in later courses build on the triangle, isosceles and exterior-angle rules.
• Constructions and bearings use the line and around-a-point facts to set directions.
• Real-world design — architecture, engineering, navigation, and computer graphics — all rest on angle reasoning.

If a later geometry topic feels tricky, the fix is almost always to revisit angle facts. Keep this revision close to hand and the multi-step problems will keep getting faster.