Angles, Parallel Lines and Polygons
Angle questions are worth marks in two parts: the number and the reason. You will be asked to “give a reason for each step”, and the reasons have specific names that examiners expect to see. Learning the names is as important as learning the rules.
1. Basic angle facts
| Rule | Sum | Reason to write |
|---|---|---|
| Angles on a straight line | 180° | “angles on a straight line” |
| Angles around a point | 360° | “angles at a point” |
| Vertically opposite angles | equal | ”vertically opposite angles” |
| Angles in a triangle | 180° | “angle sum of a triangle” |
| Angles in a quadrilateral | 360° | “angle sum of a quadrilateral” |
Complementary angles add to 90°; supplementary angles add to 180°. These two get swapped constantly — comPlementary is the smaller pair (90°), Supplementary is the Straight line (180°).
A quadrilateral sums to 360°, not 180°. Subtracting from the wrong total was a recorded error; so was subtracting from 360° in a triangle question.
2. Parallel lines
When a straight line (a transversal) crosses two parallel lines, three named relationships appear. Use the right name — this is where reasons are won and lost.
| Type | Shape | Relationship | Reason to write |
|---|---|---|---|
| Corresponding | F shape | equal | ”corresponding angles” |
| Alternate | Z shape | equal | ”alternate angles” |
| Co-interior (allied) | C or U shape | add to 180° | “co-interior angles” |
Co-interior angles are the only ones that are supplementary, not equal. Assuming all three types are equal is the most frequent parallel-lines error. If the two angles are on the same side of the transversal, between the parallel lines, they add to 180°.
“Alternate” and “vertically opposite” are different things, and calling one by the other’s name loses the reason mark even when the number is right. Alternate angles sit on opposite sides of the transversal, in a Z; vertically opposite angles are formed by two crossing lines at a single point.
These rules need parallel lines. They only apply when the lines are marked parallel (with arrows) or stated to be. Without that, none of them holds.
3. Triangles
| Triangle | Sides | Angles |
|---|---|---|
| Equilateral | all 3 equal | all 60° |
| Isosceles | 2 equal | 2 equal base angles |
| Scalene | all different | all different |
| Right-angled | — | one 90° |
In an isosceles triangle the equal angles are opposite the equal sides. Identifying the wrong pair is a common error. Mark the equal sides first, then the angles facing them.
Don’t assume a triangle is isosceles because it looks like one. You need equal sides marked, two equal angles given, or a reason such as “both are radii”. Assuming without justification was recorded repeatedly.
The exterior angle of a triangle equals the sum of the two opposite interior angles — a useful shortcut, and a valid reason in its own right.
Quadrilaterals worth knowing:
- Parallelogram — opposite sides parallel and equal; opposite angles equal; co-interior angles between the parallel sides add to 180°
- Rhombus — a parallelogram with all sides equal; diagonals bisect at 90°
- Trapezium — exactly one pair of parallel sides
- Kite — two pairs of adjacent equal sides; one pair of equal angles
Opposite angles of a parallelogram are equal — not all four. Assuming all angles are the same makes it a rectangle. Adjacent angles in a parallelogram are supplementary.
4. Polygons
Interior angles
Sum of interior angles = (n − 2) × 180°
where n is the number of sides.
| Polygon | n | Sum |
|---|---|---|
| Triangle | 3 | 180° |
| Quadrilateral | 4 | 360° |
| Pentagon | 5 | 540° |
| Hexagon | 6 | 720° |
| Heptagon | 7 | 900° |
| Octagon | 8 | 1080° |
| Decagon | 10 | 1440° |
A pentagon has 5 sides, a hexagon 6. Mixing these up was a documented error and it wrecks the whole calculation. Hex = 6, as in hexadecimal.
For a REGULAR polygon (all sides and angles equal):
Each interior angle = (n − 2) × 180° ÷ n
For a regular hexagon: 720 ÷ 6 = 120°
“Sum of the interior angles” and “one interior angle” are different questions. Giving 720° when asked for a single angle of a regular hexagon — or 120° when asked for the sum — was one of the most frequent slips in this topic. Read which is wanted.
Exterior angles
The exterior angles of ANY polygon sum to 360°.
For a regular polygon:
Each exterior angle = 360° ÷ n Number of sides n = 360° ÷ exterior angle
Interior + exterior = 180° at each vertex, because they sit on a straight line.
This is usually the faster route. For a regular polygon with interior angle 150°:
- exterior = 180 − 150 = 30°
- n = 360 ÷ 30 = 12 sides
The 360° total is for exterior angles only, and holds for every polygon regardless of the number of sides. The interior sum changes with n — mixing these two up is the core confusion in polygon questions.
5. Method — how to answer angle questions
- Mark everything given on the diagram, including equal sides and parallel arrows
- Look for the shapes: F, Z, C for parallel lines; triangles; straight lines
- Work in small steps, writing the reason each time
- Check your angles are consistent with the totals
Write a reason for every step, using the correct name. For multi-mark questions the reasons carry marks of their own. “Alternate angles”, “angles on a straight line”, “angle sum of a triangle” — the exact phrase is what earns it.
Never measure the diagram. Diagrams are not to scale. An angle is only what you can prove.
Read the question twice. Tutors made this point repeatedly here — most errors were answering for the wrong angle, not faulty arithmetic.
6. Mistakes that cost marks
Giving the angle with no reason, or with a vague one.
Calling alternate angles “opposite” or mixing up the three parallel-line names.
Treating co-interior angles as equal instead of supplementary.
Using parallel-line rules without parallel lines.
Swapping complementary and supplementary.
Using 180° for a quadrilateral, or 360° for a triangle.
Choosing the wrong pair of equal angles in an isosceles triangle.
Assuming isosceles without justification.
Assuming all angles in a parallelogram are equal.
Confusing the interior angle SUM with ONE interior angle.
Mixing up pentagon and hexagon.
Applying the 360° exterior rule to interior angles.
Measuring the diagram instead of reasoning.
Frequently asked questions
What is the sum of angles in a triangle? 180°. In a quadrilateral, 360°.
What are alternate angles? Equal angles on opposite sides of a transversal between parallel lines — the Z shape.
What are corresponding angles? Equal angles in the same position at each intersection — the F shape.
What are co-interior angles? Angles on the same side of the transversal between the parallel lines. They add to 180°.
What is the formula for the sum of interior angles? (n − 2) × 180°.
How do I find one interior angle of a regular polygon? (n − 2) × 180° ÷ n, or subtract the exterior angle from 180°.
What do exterior angles add up to? 360°, for any polygon.
How do I find the number of sides from an exterior angle? n = 360° ÷ exterior angle.
What’s the relationship between interior and exterior angles? They are on a straight line, so they add to 180°.
Do I need to give reasons? Yes — reasons carry marks, and the correct names are expected.
Quick revision checklist
- I know the basic sums: line 180°, point 360°, triangle 180°, quadrilateral 360°
- I can tell complementary from supplementary
- I know vertically opposite angles are equal
- I can name and use corresponding, alternate and co-interior angles
- I know co-interior angles add to 180°
- I check the lines really are parallel first
- I know the triangle types and their angle properties
- I pick the correct equal angles in an isosceles triangle
- I know the properties of parallelograms, rhombuses, trapezia and kites
- I can use (n − 2) × 180° for the interior sum
- I can find one interior angle of a regular polygon
- I know exterior angles always total 360°
- I can find the number of sides from an exterior angle
- I write a named reason at every step
- I never measure the diagram
These notes cover angles, parallel lines and polygons in the Cambridge IGCSE Mathematics (0580) syllabus, and are written for Grade 9–11 / Year 10–11 students. They are based on teaching patterns observed across a large set of one-to-one IGCSE Maths lessons, with particular attention to the errors students make most often and the wording examiners reward. Always check the current syllabus and formula list for your own exam series.
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