Honors Geometry Companion Book, Volume 1

1.1.3 Measuring and Constructing Angles (continued)

The third method for naming an angle is with its vertex and a point on each side, where the vertex is listed between the two points. This method is used when the angle’s vertex is either shared with another angle or it is a point on the side of another angle. D is the vertex of three angles in the given figure (one large angle and two smaller angles). Since D is the vertex of more than one angle, none of these three angles can be named simply ∠ D . And since none of these angles is labeled with a number, the third method for naming angles must be used. So, each of the three angles must be named with a point from each side and the vertex, D . To name the largest angle, first identify the sides of this angle. The sides are formed by rays DA and DC . So, A and C are each points on one of the angle’s sides. And since the vertex is D , the largest angle can be named either ∠ ADC or ∠ CDA .

Example 2 Measuring and Classifying Angles An angle is measured by relating the angle to a circle that is divided into 360 parts, shaped like pie pieces. Each one of these 360 pieces is called a degree. Imagine that an angle’s vertex is the center of a circle and that the circle itself intersects each side of the angle. The number of pie pieces, or degrees, that would be in the interior of the angle (i.e., between the two sides) is equal to the measure of the angle.

By the Protractor Postulate, a protractor can be used to find the measure of an angle that is between 0° and 180°.


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