Angle facts and parallel lines
Types of angle
| Angle | Size |
|---|---|
| less than 90° | |
| exactly 90° | |
| between 90° and 180° | |
| Straight line | exactly 180° |
| between 180° and 360° |
An angle is named by three letters with the corner in the middle: angle ABC is the angle at B, between the lines BA and BC.
Example: name the type of each angle: 35°, 90°, 127° and 245°.
- 35° is less than 90°: acute
- 90°: right angle
- 127° is between 90° and 180°: obtuse
- 245° is more than 180°: reflex
Key terms
Angle facts
Learn these facts and their exact wording:
- Angles on a straight line add up to 180°.
- Angles around a point add up to 360°.
- angles are equal (where two straight lines cross).
- Angles in a triangle add up to 180°.
- Angles in a quadrilateral add up to 360°.
- Base angles of an isosceles triangle are equal.
Example: a triangle has angles of 47° and 68°. Find the third angle.
- Angles in a triangle add up to 180°
- 180 − 47 − 68 = 65°
Example: an isosceles triangle has a top angle of 40°. Find the two base angles.
- 180 − 40 = 140
- The base angles are equal: 140 ÷ 2 = 70° each
Key terms
Angles in parallel lines
When a straight line (a transversal) crosses two parallel lines, it makes three kinds of angle pairs:
- angles are equal: they're in the same position at each crossing (an F shape).
- angles are equal: they're on opposite sides of the transversal, between the parallel lines (a Z shape).
- angles add up to 180°: they're on the same side, between the parallel lines (a C shape).
Example: a transversal crosses two parallel lines. One angle between the parallel lines is 65°. Find the alternate angle and the co-interior angle to it.
- Alternate angles are equal: 65°
- Co-interior angles add up to 180°: 180 − 65 = 115°
Exam tip
Write the reasons in full: "alternate angles are equal", not "Z angles". F, Z and C don't get marks.
Key terms
Multi-step angle problems
Most questions need several angle facts in a row. Find any angle you can, write its reason, and keep going.
- Mark the angles you know.
- Find another angle using one fact, and write the reason.
- Repeat until you reach the angle asked for.
Example: triangle ABC is isosceles, with AB = AC and angle BAC = 50°. The line BC is extended past C to a point D. Find angle ACD, giving reasons.
- Angle ABC = angle ACB, because base angles of an isosceles triangle are equal
- 180 − 50 = 130, and 130 ÷ 2 = 65, so angle ACB = 65°, because angles in a triangle add up to 180°
- Angle ACD = 180 − 65 = 115°, because angles on a straight line add up to 180°