Angle Rules

Grade 3

The basic angle facts

A few simple rules cover most angle questions at GCSE. They are not complicated — you just need to know them by name so you can quote them in proofs.

Angles on a straight line add up to 180°. Angles around a point add up to 360°. Vertically opposite angles are equal — they are the pair of angles formed across the crossing point when two lines intersect.

Every time you state an angle in a proof, give the reason in brackets. "Angles on a straight line (180°)" earns the mark — just writing 130° does not.

Angles in triangles and quadrilaterals

The angles inside any triangle add up to 180°. The angles inside any quadrilateral add up to 360°. These follow directly from the straight-line rule once you draw a diagonal.

Special triangles worth knowing: an equilateral triangle has three 60° angles; an isosceles triangle has two equal base angles.

Worked example

EXAMPLE

In triangle PQR, angle P = 47° and angle Q = 65°. Find angle R.

Angles in a triangle sum to 180°. So angle R = 180° − 47° − 65° = 68°.

Don't mix up the rules

Angles on a line = 180°, angles around a point = 360°. It is easy to use the wrong one when a diagram shows multiple lines meeting. Count carefully: is the arc showing half a turn or a full one?

Try this one

The diagram shows two parallel lines and a transversal. Angle p is marked.

Lines \( m \) and \( n \) are parallel. The angle at line \( m \) (below-right of the intersection) is \( 72^\circ \). Find angle \( p \) at line \( n \) (above-left of the intersection). [2]

Nova's hint:

This involves alternate angles formed by a transversal crossing two parallel lines.

Ask yourself: 'Are these angles on opposite sides of the transversal, between the parallel lines?' If two angles form a 'Z' or 'N' shape, what is the relationship between them?

Try to Solve it with Nova?