Angles

An angle measures turn, not length. Two angles drawn with very different arm lengths can be exactly the same angle.

An angle is the amount of turn between two rays that share an endpoint. The length of the rays has nothing to do with the size of the angle, which is the single most useful thing to establish early, because drawings are usually misleading on exactly this point.

Naming angles by size

Angle names
Name	Size	Where you see it
Acute	less than 90 degrees	The point of a slice of pizza
Right	exactly 90 degrees	The corner of a book
Obtuse	between 90 and 180 degrees	A reclined chair back
Straight	exactly 180 degrees	A flat line
Reflex	between 180 and 360 degrees	The larger way round a narrow angle

The same four kinds of angle, drawn with arms of different lengths to show that arm length does not change the measure.

Reading a protractor

A protractor carries two scales running in opposite directions, and choosing the wrong one is the standard error. The check takes a second: decide first whether the angle is acute or obtuse. If it looks acute, the answer must be under 90. If your reading says 130 for an angle that is clearly narrower than a corner, you read the wrong scale, and the right answer is 180 - 130 = 50.

Angle facts worth knowing

Angles on a straight line add to 180 degrees.
Angles around a point add to 360 degrees.
The three angles of any triangle add to 180 degrees.
The four angles of any quadrilateral add to 360 degrees.

Finding a missing angle on a straight line

two angles on a straight line one measures 118 degrees 180 - 118 = 62 degrees

Common mistakes

Judging by arm length

A 40 degree angle drawn with long arms is still 40 degrees. Longer arms make an angle look bigger and it is not.

Using the wrong protractor scale

Line the protractor centre on the vertex, one arm along the zero line, and then read the scale that starts at zero on that arm.

Check yourself

Two angles on a straight line, one is 47 degrees. What is the other?

180 - 47 = 133 degrees, which is obtuse.

A triangle has angles of 90 and 35 degrees. What is the third?

180 - 90 - 35 = 55 degrees.

Three angles meet at a point measuring 120, 95 and one unknown. Find it.

360 - 120 - 95 = 145 degrees.

Is an angle of 90 degrees acute or right?

Right. Acute means strictly less than 90.

Sources

Angle classification and basic angle facts as taught in United States elementary geometry. Diagrams are drawn for this site.