Elementary Geometry for College Students (6th Edition)

Published by Brooks Cole
ISBN 10:
ISBN 13:

Chapter 6 - Section 6.1 - Circles and Related Segments and Angles - Exercises - Page 277: 42

Answer

To do this problem, we can do a similar concept to that in problem 41, generalizing the problem to all cases of theorem 6.1.2. We are being asked to prove that the measure of the inscribed angle is equal to half of the measure of the arc that it corresponds to. In order to do this, we will consider the case at which the triangle approaches being the arc for the entire circle. When the triangle approaches being the arc for the entire circle, it approaches changing from a triangle to a straight line, meaning that the inscribed angle is approaching 180 degrees. However, the measure of the angles in the entire circle is approaching 360 degrees, since that is the measure of the angle of the whole circle. From this, it follows that the inscribed angle is $\frac{180}{36} = 1/2$ of the measure of it corresponding arc.
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