10.3 Inscribed Angles
Notes
An inscribed angle has its vertex on the circle and sides that are chords.
- Inscribed Angle Theorem: an inscribed angle is half its intercepted arc.
- Inscribed angles that intercept the same arc are congruent.
- An angle inscribed in a semicircle is a right angle.
- A quadrilateral inscribed in a circle has opposite angles supplementary.
- Example: an inscribed angle intercepts an $80^\circ$ arc, so it measures $40^\circ$.
Practice
An inscribed angle intercepts an arc of $124^\circ$. What is the angle?
Half of 124 is 62.
Triangle $ABC$ is inscribed in a circle, and $\overline{AB}$ is a diameter. What is $m\angle C$?
An angle inscribed in a semicircle is a right angle.
Quadrilateral $ABCD$ is inscribed in a circle with $m\angle A = 3x + 5$ and $m\angle C = 2x + 25$. Find $m\angle A$.
Opposite angles are supplementary: $5x + 30 = 180$, so $x = 30$ and $m\angle A = 95^\circ$.
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