Angle facts: angles on a line and around a point, vertically opposite angles, angles in parallel lines (alternate, corresponding and co-interior), angles in triangles and quadrilaterals, and the interior and exterior angles of polygons.

What this dot point is asking

Edexcel expects you to use angle facts confidently and, on "give reasons" questions, to name the rule you used. The facts cover angles on a line and around a point, vertically opposite angles, the three parallel-line relationships, angles in triangles and quadrilaterals, and the interior and exterior angles of polygons. Stating reasons precisely is as important as the arithmetic, because mark schemes award marks for the named rule.

Angle facts on lines and at points

These are the building blocks for every angle problem.

Each of these has a standard "reason" phrase that Edexcel expects, such as "angles on a straight line sum to $180^\circ$". Learning the phrases word for word secures the reasoning marks.

Angles in parallel lines

When a straight line (a transversal) crosses two parallel lines, three relationships appear.

Spotting the Z, F and C shapes is the quick way to choose the right rule, but you must still name the rule properly in your answer, not just describe the shape.

Angles in triangles and quadrilaterals

The angles of a triangle add to $180^\circ$; the angles of a quadrilateral add to $360^\circ$. A useful related fact is that an exterior angle of a triangle equals the sum of the two interior angles at the other two vertices, which often saves a step.

Interior and exterior angles of polygons

A polygon's angles follow two key rules, one for the interior angles and one for the exterior.

The exterior-angle route is usually fastest for regular polygons, while the $(n - 2) \times 180^\circ$ formula is needed when the polygon is irregular and you know all but one angle.

Working with reasons on multi-step problems

Harder angle questions combine several facts in one diagram, and Edexcel awards marks for each correctly reasoned step. The method is to find every angle you can immediately, writing the reason next to each, then use those to find the rest. For instance, you might first use "base angles of an isosceles triangle are equal", then "angles in a triangle sum to $180^\circ$", then "alternate angles are equal" to carry a value across to another part of the diagram. Laying the reasons out in order shows a logical chain, which is exactly what the "give reasons" mark scheme rewards. A value with no reason scores only part marks even when it is correct.

Try this

Q1. The interior angles of a polygon sum to $1080^\circ$. How many sides does it have? [2 marks]

• Cue. $(n - 2) \times 180 = 1080$, so $n - 2 = 6$ and $n = 8$.

Q2. Four angles meet at a point. Three of them are $90^\circ$, $120^\circ$ and $65^\circ$. Work out the fourth. [2 marks]

• Cue. Angles around a point sum to $360^\circ$: $360 - (90 + 120 + 65) = 85^\circ$.