Angles and angle facts

Find missing angles using facts about straight lines, points, triangles and parallel lines, and give a reason for each step.

About 3 min read
  • Use angle facts on straight lines and around a point
  • Use angles in a triangle and vertically opposite angles
  • Use alternate and corresponding angles on parallel lines
  • Give a geometric reason for each step
Before you start

You can measure angles in degrees and solve simple equations. Both skills are used to find missing angles.

Facts about angles

Angles follow fixed rules, so you can often calculate an angle without measuring it. Angles on a straight line add up to 180°, and angles around a point add up to 360°. When two straight lines cross, the opposite angles are equal. The angles inside any triangle add up to 180°.

Definition: vertically opposite angles

The equal angles on opposite sides of the point where two straight lines cross.

Parallel lines

Parallel lines never meet. When another line crosses them, it creates pairs of related angles. Corresponding angles are equal and form an F shape. Alternate angles are equal and form a Z shape. Co-interior angles add up to 180° and form a C shape.

Definition: alternate angles

Equal angles on opposite sides of a line crossing two parallel lines.

Find a missing angle

A triangle has angles of 52° and 71°. Find the third angle, then the angle next to it on a straight line.

Maths
third angle = 180 − 52 − 71 = 57°     angles in a triangle add to 180°
next angle = 180 − 57 = 123°          angles on a straight line add to 180°

Answer: 57° and 123°.

Diagrams can mislead

Diagrams are often not drawn to scale. Work out each angle from the facts instead of judging by eye.

Exam focus
Say it like this“x = 65° because alternate angles are equal.”
Not like this“x = 65° because of the Z shape.” — Use the correct geometric name for the reason.
Common mistakeGiving the right angle but no reason when the question asks you to give reasons.

Practice

Two angles on a straight line are 110° and x. Find x.
x = 180° − 110° = 70°, because angles on a straight line add to 180°.
Three angles around a point are 120°, 95° and y. Find y.
y = 360° − 120° − 95° = 145°, because angles around a point add to 360°.
An isosceles triangle has a top angle of 40°. Find each base angle.
The base angles are equal: (180° − 40°) ÷ 2 = 70°.
Going further
Two angles on a straight line are 2x + 10 and 3x − 5. Find x.
2x + 10 + 3x − 5 = 180, so 5x + 5 = 180, so 5x = 175 and x = 35.

Quick check

Angle facts quick check

What do the angles in a triangle add up to?

Any triangle's three angles fit together to make a straight line.

Which pair of angles on parallel lines is equal and forms a Z shape?

They sit on opposite sides of the crossing line, between the parallel lines.

Two lines cross and one angle is 48°. What is the opposite angle?

Vertically opposite angles are always equal, so the size stays the same.
Key takeaways
  • Straight line angles add to 180° and angles around a point add to 360°.
  • Angles in a triangle add to 180°.
  • On parallel lines, alternate and corresponding angles are equal.
  • Give a reason for each step.
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