Angle Rules on Lines and Around a Point

What you will learn

• Use angles on a straight line and around a point.
• Use vertically opposite angles.
• Use angles in parallel lines.
• Give a reason for each step.

Before you start: you should know the types of angle and the size of a full and a half turn. See Angles.

The idea

A few simple facts let you find almost any missing angle.

Angles on a straight line

A straight line is a half turn, so the angles on it add to $180^\circ$.

Angles around a point

All the angles that meet at a point make a full turn, so they add to $360^\circ$.

Vertically opposite angles

When two straight lines cross they make four angles. The ones facing each other are equal.

Parallel lines

When a straight line (a transversal) crosses two parallel lines, it makes eight angles but only two different sizes.

• Corresponding angles (in matching positions, an F-shape) are equal.
• Alternate angles (on opposite sides of the line, a Z-shape) are equal.
• Co-interior angles (between the parallels on the same side, a C-shape) add to $180^\circ$.

Worked examples

Example 1: Angles around a point

Three angles meet at a point: $110^\circ$, $2x$ and $3x$. Find $x$.

1. Angles around a point sum to $360^\circ$: $110 + 2x + 3x = 360$.
2. Simplify: $5x = 250$.
3. Divide: $x = 50$.
4. Check: $110 + 100 + 150 = 360$.

Answer: $x = 50^\circ$

Example 2: Parallel lines

A transversal crosses two parallel lines. One angle is $72^\circ$. Find its co-interior partner $y$ and the alternate angle $z$.

1. Co-interior angles add to $180^\circ$: $y = 180^\circ - 72^\circ = 108^\circ$.
2. Alternate angles are equal: $z = 72^\circ$.

Answer: $y = 108^\circ$, $z = 72^\circ$

Example 3: Crossing lines

Two lines cross and one angle is $35^\circ$. Find the other three.

1. The opposite angle is also $35^\circ$.
2. The neighbours are on a straight line with it: $180 - 35 = 145^\circ$, and the last one is also $145^\circ$.
3. Check: $35 + 145 + 35 + 145 = 360$.

Answer: $35^\circ$, $145^\circ$, $35^\circ$, $145^\circ$

Tip: when you give a reason, name the fact you used, such as “angles on a straight line”. In exams the reason is often worth as much as the number.

Why it matters

Angle facts are the starting point for triangles, polygons and circle theorems. They are used by surveyors, builders and designers to work out angles they cannot measure directly.