Types of angles: acute, right, obtuse, straight, reflex and a full turn, with the degree range of each, how to name an angle with three letters, how to measure one with a protractor (and the baseline mistake), complementary and supplementary pairs, the angle facts on a line, at a point and in a triangle, and a printable chart

An angle is the amount of turn between two lines that meet at a point, measured in degrees, with 360 degrees in a full turn. Every angle in school geometry is one of six types, decided only by its size, and "obtuse angle", "acute angle" and "reflex angle" are searched hundreds of thousands of times a month because the names are learned in Grade 4 and tested until Grade 10. The chart, the degree ranges, the protractor method and the angle facts are below, with the printable.

Download the types of angles chart and practice sheet (PDF) (the six types with the degree ranges on page 1, the angle facts, complementary and supplementary pairs and a practice set with answers on page 2; two pages, free).

The six types of angles

Two facts the mark schemes test. 90 is not obtuse and 180 is not reflex: the boundary values have their own names, and an answer of "obtuse" for a right angle is wrong, not nearly right. Every angle has a reflex partner: the 60° angle between two lines sits beside a 300° angle on the other side, because the two together make the full turn of 360°. Which one the question means is shown by the arc; with no arc, the smaller angle is meant.

Naming an angle

An angle is named by three letters, with the vertex (the corner) in the middle: angle ABC, written ∠ABC, is the angle at B between the arms BA and BC. ∠ABC and ∠CBA are the same angle; ∠BAC is a different one, at A. When only one angle sits at a vertex, the single letter is enough (∠B), and in diagrams a lower-case letter or a number inside the arc (a, x, 1) names it too.

How to measure with a protractor

1. Decide the type first. Look at the angle: acute (less than a square corner) or obtuse (more). Write it down. This is the check for step 4.
2. Centre on the vertex. The small hole or cross at the middle of the protractor's straight edge goes exactly on the corner of the angle.
3. Baseline on one arm. Turn the protractor so its 0 line lies along one arm of the angle. Extend the arms with a ruler if they are too short to reach the scale.
4. Read the right scale. A protractor has two scales, one running 0 to 180 from the left and one from the right. Use the scale that reads 0 on the arm you lined up, and read where the other arm crosses. The number must agree with step 1: if you wrote "acute" and the scale says 130, you read the wrong scale, and the answer is 50.
5. Reflex angles. A half-circle protractor only goes to 180. Measure the smaller angle on the other side and subtract from 360: if the inside angle is 70°, the reflex angle is 360 − 70 = 290°.

The baseline mistake, reading the wrong scale, is the single most common error in angle questions from Grade 4 to Grade 8, and step 1 removes it.

Pairs of angles

The angle facts used in every problem

• Angles on a straight line add to 180°. Two angles on a line, one of them 115°: the other is 65°.
• Angles at a point add to 360°. Three angles round a point, 120°, 150° and x: x = 360 − 270 = 90°.
• Angles in a triangle add to 180°. A triangle with 40° and 75°: the third angle is 65°. An obtuse triangle has one obtuse angle and two acute; a right triangle has one right angle and two acute that are complementary; a triangle cannot have two obtuse angles, because two angles over 90 already pass 180.
• Vertically opposite angles are equal. Where two straight lines cross, the four angles come in two equal pairs, and each pair of neighbours is supplementary.
• Angles in a quadrilateral add to 360°, because a diagonal cuts it into two triangles.

Worked examples

1. Classify 137°. Between 90 and 180: obtuse.
2. Classify 90°. Exactly 90: right, not acute and not obtuse.
3. Classify 270°. Between 180 and 360: reflex. It is three-quarters of a turn.
4. Two angles on a straight line are x and 2x. Find x. x + 2x = 180, so 3x = 180 and x = 60°; the angles are 60° and 120°.
5. The complement of 28°. 90 − 28 = 62°. Its supplement is 180 − 28 = 152°.
6. A protractor reads 50 and 130 where the second arm crosses, and the angle looks acute. It is 50°.
7. A triangle has angles 90° and 35°. The third is 180 − 125 = 55°, and 35 and 55 are complementary, as the two acute angles of a right triangle always are.

The mistakes that lose marks

• Calling 90° obtuse or 180° reflex: the boundary values have their own names.
• Reading the wrong protractor scale (the 130 for a 50). Decide acute or obtuse before you read.
• Measuring from the wrong arm: the baseline must lie along one arm with 0 on it.
• Writing the vertex at the end of the name: ∠ABC has its vertex at B, the middle letter.
• Marking a right angle with an arc: it takes a small square, and a square on a diagram means exactly 90° even if the drawing looks off.
• Mixing up complementary and supplementary: complementary is 90, supplementary is 180.

Where to get help

If the method is not sticking from a page, a 1-on-1 online class fixes it faster than another worksheet: one tutor with a master's degree, one child, on a shared whiteboard, in your time zone (7 am before school, after 4 pm and weekend mornings, US time; after-school slots for the UK, Canada, Australia and the UAE), for about US$10 to $13 an hour. The first 30-minute class is free: how 1-on-1 online tutoring works.

Related pages

• Parallel lines cut by a transversal: the eight angle pairs
• Area of a sector and arc length
• The coordinate grid: how to plot points

Questions parents ask

See how we teach this, 1-on-1 online →