Angles

ANGLES

Angles are formed by two rays with a common endpoint. A pair of opposite rays not only form a line, but also form a straight angle, whose measure is 180˚. Angles also partition a plane into three regions. (1) the region of points in the interior of an angle (2) the region of points in the exterior of the angle (3) the region of points that form the angle itself.

GeoGebra Worksheet

Grab the points on the angle to move the angle, change its measure, and relocate the interior and exterior regions.

Classifying Angles

Angles are classified by their measurements. An acute angle has a measure that is greater than 0˚ but less than 90˚. A right angle has a measure exactly equal to 90˚. An obtuse angle has a measure that is greater than 90˚ but less than 180˚, and finally, a straight angle has a measure that is exactly equal to 180˚.

The math statement below is read as: "the measure of angle CAB equals 90 degrees"

Whenever a lowercase m precedes the name of a geometric figure it implies that a measurement is being given. After the lowercase m there is the symbol indicating that the figure being discussed is an angle, and then the name of the angle. The equal symbol is followed by the value of the angle measure along with a symbolic label. 90˚ is ninety degrees. Degrees is the label on the measurement. When you measure a line segment with a ruler your label may be feet, inches, meters, centimeters, etc. However, when measuring angles in geometry the label is in degrees. Angles do have other labels, but that discussion is for another subject.

Naming Angles

There are three different ways for naming angles. Angles may be named by a single point, the vertex of the angle, as long as the vertex point is unique. What does it mean to say that the vertex point is unique?

A unique vertex is a vertex for exactly one angle. If two different angles have the same vertex point, then those angles cannot be named by a single point. A second way to name angles is the three point method. Suppose we have an angle named CAB. This means that point C is on one side of the angle, point A is the vertex of the angel, and point be is on the other side of the angle. If more than one angle shares a vertex, then we use the three point method to name the angles so they can be correctly identified.

Single Vertex

When there is a single angle then it can be named using the vertex point. There is no confusion as to which angle is ⦣A.

However, you still can use the three point method for naming the angle. It is good practice to refer to the angle as ⦣CAB. Notes, that the vertex point must be the middle letter when using the three point naming technique.

Shared Vertex

When more than one angles shares the vertex point, then more specified naming conventions are necessary to avoid confusion as to which angle is the focus.

Which angle is ⦣X ?

Therefore, either the three point naming convention or numeric angle names are used.

⦣1 = ⦣WXZ & ⦣2 = ⦣ZXY

We would not refer to any angle as ⦣X.

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