theorem of inscribed angle. With proof
Answers
inscribed angle if its vertex is on the circle and its sides contain chords of the circle. How is the measure of an inscribed angle related to the measure of the corresponding central angle?
Exploring the Concept:
Construct a circle. Label its center P.
Use a straightedge to construct a central angle. Label it .
Locate three points on Circle P in the exterior of and label them T, U, and V. Use a straightedge to draw the inscribed angles ., and.
Investigate:
Use a protractor to measure , , , and. Make a table similar to the one below. Record the angle measures for Circle 1 in the table.
Repeat Steps 1 through 3 using different central angles. Record the measures in your table.
Make a Conjecture:
Use the results in your table to make a conjecture about how the measure of an inscribed angle is related to the measure of the corresponding central angle.
An inscribed angle is an angle whose vertex is on a circle and whose sides contain chords of the circle. The arc that lies in the interior of an inscribed angle and has endpoints on the angle is called the intercepted arc of the angle. Measure of an Inscribed Angle Theorem
If an angle is inscribed in a circle, then its measure is half the measure of its intercepted arc.
This proof can be split into 3 different cases. Each will differ in where the center lies in relation to the inscribed angle.
Theorem: If two inscribed angles of a circle intercept the same arc, then the angles are congruent.