You will use three corollaries to the Inscribed Angle Theorem to find measures of angles in circles. The first corollary may confirm an observation you made in the Solve It.
Take Note Corollaries to Theorem 12-11: The Inscribed Angle Theorem
Corollary 1
Two inscribed angles that intercept the same arc are congruent.
Corollary 2
An angle inscribed in a semicircle is a right angle.
Corollary 3
The opposite angles of a quadrilateral inscribed in a circle are supplementary.
You will prove these corollaries in Exercises 31–33.
Problem 2 Using Corollaries to Find Angle Measures
What is the measure of each numbered angle?
Think
Is there too much information?
Each diagram has more information than you need. Focus on what you need to find.
✓ Got It?
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In the diagram below, what is the measure of each numbered angle?
The following diagram shows point A moving along the circle until a tangent is formed. From the Inscribed Angle Theorem, you know that in the first three diagrams
Table of Contents
- 6-1 The Polygon Angle-Sum Theorems
- 6-2 Properties of Parallelograms
- 6-3 Proving That a Quadrilateral Is a Parallelogram
- 6-4 Properties of Rhombuses, Rectangles, and Squares
- 6-5 Conditions for Rhombuses, Rectangles, and Squares
- 6-6 Trapezoids and Kites
- 6-7 Polygons in the Coordinate Plane
- 6-8 and 6-9 Coordinate Geometry and Coordinate Proofs