Page 136 - IM_FL_Geometry_Print Sample

Student Lesson Summary
We have discussed the central angle formed from an arc by the two radii and . We can also explore an inscribed angle formed by connecting another point on the circle to the points and . A line segment with both endpoints on a circle is called a chord and so our inscribed angle was formed by two chords meeting at the same point. As long as we place our point on the circle outside the arc the measure of the inscribed angle will always be exactly the measure of the central angle . In particular, all inscribed angles
formed in this way on the same arc will have the same angle measure. For example, if the central angle measures degrees, the inscribed angle would measure
degrees, even as point moves all along the circumference outside of arc .
A special case of this occurs when our arc is exactly half the circle. In this case the chord forms a diameter of the circle and the measure of is radians or . It follows that
our inscribed angle is half of that, namely radians or , a right angle. The converse
of this is also true. If our inscribed angle is a right angle it follows that the central angle must be radians or and thus the arc must be half the circle and the chord must be a diameter.
Unit 7
Lesson 6: The Angle Inscribed 63