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Definitions

Individual Angles

Mathematicians categorize angles based on their measures:

An angle is acute if its measure is strictly less than 90°.

An angle is a right angle if its measure is exactly 90°.

An angle is obtuse if its measure is strictly greater than 90°.

Some textbooks call an angle whose measure is 180° a straight angle. Essentially this is a line so some books don't consider it an angle at all.

Take a look at some examples.

an acute angle
(m < 90°)
a right angle
(m = 90°)
an obtuse angle
(90° < m < 180°)
a straightangle
(m = 180°)

Note the little box in the right angle. Right angles are sort of special and come up a lot so there's a special way to let you know that's what you're looking at.

Pairs of Angles

Mathematicians also classify pairs of angles based on their angle measures and their arrangement.

Two angles are complementary if the sum of their angle measures is 90°. The angles in the picture to the right are complementary because 30° + 60° = 90°.
Two angles are supplementary if the sum of their angle measures is 180°. The angles in the picture to the right are supplementary because 60° + 120° = 180°.
Two angles are adjacent if they have a common side and no interior points in common. Another way of saying "have a common side" would be "they have the same vertex". In the diagram ∠ABD and ∠DBC are adjacent angles but ∠DBC and ∠ABC are not because they have interior points in common (e.g. E).
Vertical angles are the non-adjacent angles formed by two intersecting lines. In the diagram to the left angles 1 and 3 are vertical angles but angles 1 and 4 are not because they're adjacent. (Angles 2 and 4 are vertical also - vertical angles always come in sets of two.)
Two angles form a linear pair if they are adjacent and supplementary.