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Since the time of the ancient Greeks (about 500 B.C.E.), angles have been measured in terms of a circle. In fact, about 1,000 years earlier, the Babylonians first divided the circle into 360 equal parts for astronomical purposes, providing a convenient unit — the degree — for expressing the measure of any angle. A degree can be further divided into 60 minutes, and a minute can be divided into 60 seconds. This level of detail is frequently seen in longitude and latitude, but almost never in school mathematics.

While degrees are the most commonly used units of angle measure, there are also other units. For example, angles are sometimes measured in radians in order to simplify certain calculations. The radian measure is defined in the International System of Units (SI) as the ratio of arc length to the radius of the circle. For 1 radian, the arc length is equal to radius:

An angle can be defined as the union of two rays with a common endpoint. (A ray begins at a point and extends infinitely in one direction.) The common endpoint is called the vertex (A in the figure below), and the rays are called the sides of the angle.

It’s customary to name an angle using an angle sign () followed by three letters: one that corresponds to a point on one of the rays, a second that corresponds to the vertex, and a third that corresponds to a point on the other ray. When it’s clear from the context, though, you can just use the letter for the vertex.

For example, you could name the angle illustrated below BAC (read “angle BAC”), CAB (read “angle CAB”), or A (read “angle A”). Point A is the vertex, and rays AB and AC are the sides:

Here’s an interesting fact: The two rays that define the angle can be on the same line. There are two ways this can happen:

While these angles may look much the same in the diagram, they are quite different geometrically:

- ∠PQR is an angle whose sides are opposite rays. This type of angle is called a straight angle.
- ∠QPT is an angle whose sides (PQ and PT) are coincident. This type of angle is called a zero angle.

When we measure an angle, no matter how it is classified, what we’re measuring is the amount of turn. This raises an important question: Do the lengths of the sides of an angle in any way affect the measure of that angle?

To examine this question, make an informal protractor using bendable straws (see the picture below): **Note 2**

Use your protractor to demonstrate how angles are the result of the amount of turn by forming acute, obtuse, and right angles:

**Problem A1**

Try changing the lengths of the sides on your protractor by adding more straws or cutting the original straw.

What happens to the angle when you do this? Do the lengths of the sides of an angle affect the measure of that angle?

When you use a straw protractor to show an angle, you can see that there are in fact two angles displayed — an angle inside the rays (the interior angle) and an angle outside the rays (the reflex angle):

The measure of the reflex angle is between 180 and 360 degrees. The interior and reflex angle measures sum to 360 degrees.

**Note 2**

To make an informal protractor from bendable straws, first take one straw and make a slit in the short section all the way from the end to the bendable portion. Cut the second straw so that you only have the long section (after the bend). Finally, compress the short section of the first straw (which you cut) and fit it into one end of the second cut straw. You now have a simple protractor.

**Problem A1**

Nothing happens to the angle. The lengths of the sides of an angle do not affect the measure of that angle.