This note covers the rules for angles on straight lines, around points, between parallel lines, inside triangles and quadrilaterals, and inside polygons. These rules are the tools you use to find missing angles in almost every geometry problem, so it pays to know each one by name and by sight.
Key ideas
Basic angle facts
- Angles on a straight line add up to \(180^\circ\).
- Angles around a point add up to \(360^\circ\).
- Vertically opposite angles (the pairs formed when two straight lines cross) are equal.
- A right angle is \(90^\circ\). An acute angle is less than \(90^\circ\). An obtuse angle is between \(90^\circ\) and \(180^\circ\). A reflex angle is between \(180^\circ\) and \(360^\circ\).
Parallel lines. When a straight line (a transversal) crosses two parallel lines, it creates pairs of related angles.
- Corresponding angles are equal. They sit in the same position at each crossing, forming an F shape.
- Alternate angles are equal. They lie on opposite sides of the transversal between the parallel lines, forming a Z shape.
- Co-interior angles add up to \(180^\circ\). They lie on the same side of the transversal between the parallel lines, forming a C or U shape.
Triangles and quadrilaterals
- The angles in a triangle add up to \(180^\circ\).
- In an isosceles triangle, the two angles opposite the equal sides are equal. In an equilateral triangle, every angle is \(60^\circ\).
- An exterior angle of a triangle equals the sum of the two interior angles opposite to it.
- The angles in a quadrilateral add up to \(360^\circ\).
- Special quadrilaterals: a parallelogram has opposite angles equal. A rhombus is a parallelogram with four equal sides. A kite has one pair of equal opposite angles. A trapezium has one pair of parallel sides.
Polygons. A polygon with \(n\) sides can be split into \(n - 2\) triangles from one vertex. So the interior angles add up to \((n - 2) \times 180^\circ\). The exterior angles of any polygon add up to \(360^\circ\). Each interior angle and its exterior angle lie on a straight line, so they add up to \(180^\circ\).
A regular polygon has equal sides and equal angles. For a regular polygon with \(n\) sides, each exterior angle is \(\dfrac{360^\circ}{n}\) and each interior angle is \(180^\circ - \dfrac{360^\circ}{n}\).
Polygon names you should recognise: triangle (3), quadrilateral (4), pentagon (5), hexagon (6), heptagon (7), octagon (8), nonagon (9), decagon (10).
Whenever you use a rule in an answer, say its name, such as "alternate angles" or "angles in a triangle". This makes your reasoning clear and helps you spot mistakes.
Many problems need two or three rules in a row. A good habit is to look for the angle you can find straight away, such as one on a straight line or in an isosceles triangle, and then use it to unlock the next angle. Angles outside a polygon on a straight line with a corner can also be found by subtracting the interior angle from \(180^\circ\).
Definitions and formulae
| Term or formula | Meaning / when to use |
|---|---|
| Angles on a line | Sum to \(180^\circ\). |
| Angles at a point | Sum to \(360^\circ\). |
| Vertically opposite | Equal. |
| Corresponding / alternate | Equal, when the lines are parallel. |
| Co-interior | Sum to \(180^\circ\), when the lines are parallel. |
| Triangle angle sum | \(180^\circ\). |
| Quadrilateral angle sum | \(360^\circ\). |
| Interior angle sum of an \(n\)-sided polygon | \((n - 2) \times 180^\circ\). |
| Exterior angle sum | \(360^\circ\) for any polygon. |
| Regular polygon, exterior angle | \(\dfrac{360^\circ}{n}\). |
| Regular polygon, interior angle | \(180^\circ - \dfrac{360^\circ}{n}\), or \(\dfrac{(n-2) \times 180^\circ}{n}\). |
Worked examples
Example 1 Two parallel rails are crossed by a straight pipe. Two co-interior angles are \((2x + 15)^\circ\) and \((x + 30)^\circ\). Find \(x\) and both angles.
Step 1: co-interior angles between parallel lines add up to \(180^\circ\). So \((2x + 15) + (x + 30) = 180\).
Step 2: simplify. \(3x + 45 = 180\), so \(3x = 135\) and \(x = 45\).
Step 3: the angles are \(2(45) + 15 = 105^\circ\) and \(45 + 30 = 75^\circ\).
Step 4: check. \(105 + 75 = 180\).
Answer: \(x = 45\); the angles are \(105^\circ\) and \(75^\circ\).
Example 2 A pentagonal paving stone has interior angles \(2x^\circ\), \((x + 30)^\circ\), \(115^\circ\), \(95^\circ\) and \(120^\circ\). Find \(x\).
Step 1: the interior angles of a pentagon add up to \((5 - 2) \times 180 = 540^\circ\).
Step 2: write the equation. \(2x + (x + 30) + 115 + 95 + 120 = 540\).
Step 3: simplify. \(3x + 360 = 540\), so \(3x = 180\) and \(x = 60\).
Step 4: the angles are \(120^\circ, 90^\circ, 115^\circ, 95^\circ, 120^\circ\), which add up to \(540^\circ\).
Answer: \(x = 60\)
Example 3 Each interior angle of a regular polygon is \(156^\circ\). How many sides does it have, and what is the sum of its interior angles?
Step 1: each exterior angle is \(180^\circ - 156^\circ = 24^\circ\).
Step 2: the exterior angles add up to \(360^\circ\), so the number of sides is \(\dfrac{360}{24} = 15\).
Step 3: the interior angle sum is \((15 - 2) \times 180 = 13 \times 180 = 2340^\circ\).
Step 4: check. \(15 \times 156 = 2340\).
Answer: 15 sides; the sum of interior angles is \(2340^\circ\).
Common errors
- Using alternate or corresponding angle rules when the lines are not marked as parallel. The rules only work for parallel lines.
- Mixing up co-interior angles (sum to \(180^\circ\)) with alternate angles (equal).
- Using \(n \times 180^\circ\) instead of \((n - 2) \times 180^\circ\) for the interior angle sum.
- Dividing \(360^\circ\) by the interior angle instead of the exterior angle when finding the number of sides.
- Giving an answer without a reason. Always name the angle rule you used.
Exam tip: label every angle you find on the diagram as you go. Later steps often depend on an earlier angle, and a labelled diagram keeps your working clear if you need to find a mistake.
Check yourself
- Find each interior angle of a regular octagon.
Answer
Exterior angle \(= 360 \div 8 = 45^\circ\). Interior angle \(= 180 - 45 = 135^\circ\). - What is the sum of the interior angles of a decagon?
Answer
\((10 - 2) \times 180 = 8 \times 180 = 1440^\circ\). - A regular polygon has an exterior angle of \(20^\circ\). How many sides does it have?
Answer
\(360 \div 20 = 18\) sides. - Triangle \(ABC\) is isosceles with \(AB = AC\) and angle \(A = 40^\circ\). Side \(BC\) is extended past \(C\). Find the exterior angle at \(C\).
Answer
The base angles are \((180 - 40) \div 2 = 70^\circ\). The exterior angle at \(C\) is \(180 - 70 = 110^\circ\). This also equals \(40 + 70\), the two opposite interior angles.