degree measure
When you turn all the way around once and face the same way again, you have made a full turn. The ancient Babylonians chose to slice that full turn into 360 equal pieces, and each piece is one degree, written with a small circle, as in 90°. Degree measure puts a number on how wide an angle is.
One full revolution is 360 degrees, a half-turn (a straight angle) is 180 degrees, and a quarter-turn (a right angle) is 90 degrees. We read an angle's degree measure with a protractor: place its centre on the vertex, align the zero line along one side of the angle, and read the number where the other side crosses the scale. The measure of angle ABC is written m(angle ABC) = 50°, say. The number 360 was likely chosen because it is close to the days in a year and divides evenly by so many numbers (2, 3, 4, 5, 6, 8, 9, 10, 12, ...), which makes common fractions of a turn land on whole degrees.
Degrees are not the only way to measure angles: in higher mathematics the radian is preferred because it makes the calculus of trigonometric functions clean, with a full turn equal to 2·pi radians. But degrees remain the everyday unit, and the conversion is simply 180 degrees = pi radians. A common slip is to forget that an angle's measure is independent of how big you draw it — a 30-degree angle is 30 degrees whether its rays are an inch or a mile long.
To measure angle ABC with a protractor, set the centre on vertex B and the 0° line along ray BA; if ray BC crosses the scale at 65, then m(angle ABC) = 65°. Splitting that angle in half with a bisector gives two angles of 32.5° each.
A protractor reads the rotation between the two sides; the answer is a number of degrees.
An angle's measure depends only on the rotation between its sides, never on how long the sides are drawn; degrees and radians are two units for the same thing, with 180° = pi radians.