Geometry: Constructions

Angles

An angle is a geometric figure consisting of two rays with a common endpoint. It looks like this:

Figure %: Angle ABC
The common endpoint is called the vertex of the angle; in this case the vertex is point A, which is a part of the ray AB as well as the ray AC. The angle can be called either angle CAB or angle BAC. The only rule in naming an angle is that the vertex must always be the middle "initial" of the angle. The symbol for an angle is this:

Figure %: The symbol for angle ABC

Measuring Angles

Long ago people wanted to measure angles, so numbers were arbitrarily assigned to determine the size of angles. Under this arbitrary numbering system, one complete rotation around a point is equal to a 360 degree rotation. (There is another unit of measure for angles besides degrees called radians, in which one full rotation is equal to 2Π radians; in this text we will use degrees as our default unit for measuring angles.) Two angles with the same measure are called congruent angles. Congruence in angles is symbolized by a small arc drawn in the region between rays. Congruent angles are drawn with the same number of such arcs between their rays. An angle's measure determines how it is classified.

Zero Angles

An angle with a measure of zero degrees is called a zero angle. If this is hard to visualize, consider two rays that form some angle greater than zero degrees, like the rays in the . Then picture one of the rays rotating toward the other ray until they both lie in the same line. The angle they create has been shrunk from its original measure to zero degrees. The angle that is now formed has a measure of zero degrees.

Figure %: A zero angle

Right Angles

An angle with a measure of 90 degrees is called a right angle. A right angle is symbolized with a square drawn in the corner of the angle.

Figure %: A right angle

Straight Angles

An angle with a measure of 180 degrees is called a straight angle. It looks just like a line. Don't mix up straight angles with zero angles.

Figure %: A straight angle

Acute and Obtuse angles

Another way to classify angles by their measures is to consider whether the angle's measure is greater or less than 90 degrees. If an angle measures less than 90 degrees, it is called an acute angle. If it measures more than 90 degrees, it is called an obtuse angle. Right angles are neither acute nor obtuse. They're just right.

Figure %: Acute and obtuse angles

Interior and Exterior Angles

So far, all of the angles we have looked at and studied have been interior angles. When two rays share a common endpoint, two angles are created. Up until now, we have only studied the interior angle: the angle whose measure is less than 180 degrees. But actually, whenever two rays create an angle of less than 180 degrees, they also create another angle whose measure is 360 degrees minus the measure of the smaller angle. As we said before, the smaller angle, whose measure is less than 180 degrees, is the interior angle. The other angle, which seems to rotate around the "outside" of the interior angle, is the exterior angle. The measure of the exterior angle is always greater than that of the interior angle, and is always equal to 360 degrees minus the measure of the interior angle. Below both are pictured.

Figure %: An Interior and Exterior Angle

In the following sections, we'll study pairs of angles and relationships between angles. In these sections, it will be important to understand properties of angles that lie next to each other. Formally, these angles are called adjacent angles. Three things must be true for angles to be adjacent:

1. The two angles must share a common vertex.
2. They must share one common side.
3. The angles must not share any interior points.
See how each statement is true for the adjacent angles below.
Angle CAB is adjacent to angle DAB. The angles share a common vertex, A, a common side, ray AB, and share no interior points (the ray AB is not on the interior of either angle, it only forms a side of each).

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