Another axiom, a generalization of Euclid's fourth postulate, says the measure of any angle does not change if you move it around in the plane. Two angles with the same measure are called congruent.
The angle abc, determined by the points a b and c, consists of the ray ba and the ray bc. Thus b is at the apex of the angle. Think of a and c as numbers on the clock, and b is where the hands meet. The hands indicate the two rays. (We're not talking about a digital clock here.) The symbol ∠ is sometimes used to denote an angle, as in ∠abc.
Angles are assigned numbers, in degrees. The angle aba is the trivial angle, one ray, and measures 0°.
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If ∠bac and ∠cad are angles, sharing a common ray ac,
with ray ac inside the larger angle bad,
the measure of ∠bad is the measure of ∠bac + the measure of ∠cad.
This is an axiom, called the angle addition postulate.
Another axiom, a generalization of Euclid's fourth postulate, says the measure of any angle does not change if you move it around in the plane. Two angles with the same measure are called congruent. |
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Finally, angles that wrap all the way around a central point add up to 360°. Let's put all this together, using the clock as an illustration. Point one hand at 12, straight up, and the other at 1. This is an angle, x°. If the hands are locked together and moved to point to 1 and 2, adjacent to the previous angle, the measure of the angle between 12 and 2 is 2x°. Repeat this process 12 times around and 12x = 360. Therefore the angle between 12 oclock and 1 oclock is 30°.
An angle that is a straight line, two rays pointed in opposite directions, measures 180°, and is called the straight angle. A right angle measures 90°, and two right angles make a straight angle. An acute angle is more than 0° and less than 90°, and an obtuse angle is more than 90° and less than 180°.
| Complementary angles add up to 90°. Supplementary angles add up to 180°. Note that two supplementary angles are both right angles, or one is acute and the other is obtuse. |
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An angle bisector is a ray that cuts the angle in half. Bisect a right angle and get two angles of 45°. An angle is trisected when it is cut by two rays into 3 equal pieces.