Prove the theorem for the case where one side of the inscribed angle passes through the center O.
This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
The theorem states that the measure of an angle inscribed in a circle is half the measure of the arc it subtends. Let $\angle ABC$ be an inscribed angle subtending arc $AC$ in a circle with center $O$.
Step 1: Prove the theorem for the case where one side of the inscribed angle passes through the center $O$.
Assume side $BC$ is a diameter. Draw radius $OA$.
In $\triangle OAB$, $OA = OB$ because they are both radii of the same circle, making $\triangle OAB$ an isosceles triangle.
Therefore, the base angles are equal: $m(\angle OAB) = m(\angle OBA)$.
The central angle $\angle AOC$ is an exterior angle to $\triangle OAB$.
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