Inscribed angles are useful to proving certain characteristic of a circle. Memorizing the formulas of an inscribed angle will be useful in geometry.
Inscribed Angle: An angle where the vertex is on the circle’s circumference and the sides contain chords.
Intercepted Arc: The arc that is on the interior of the inscribed angle and whose endpoints are on the angle.
Inscribed Polygon: A polygon inside the circle with all the vertices on the circle’s circumference.
Circumscribed Circle: A circle that contains all the vertices of a polygon.
The vertex of an inscribed angle can be anywhere on the circle as long as its sides intersect the circle. The arc that is formed on the interior of the inscribed angle is the intercepted arc.
The center of the circle can be on the side of the inscribed angle, or it can be either inside or outside the angle.
Theorem: The measure of an inscribed angle is half the measure of its intercepted arc.
Theorem: Inscribed angles that intercept the same arc are congruent.
Theorem: An angle that intercepts a semicircle is a right angle.
When all the vertices of a polygon lie on a circle, the polygon is an inscribed polygon. The circle containing the vertices is a circumscribed circle.
Theorem: If a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle.
Not all quadrilaterals can be inscribed in a circle. Quadrilaterals that can be inscribed in a circle are called cyclic quadrilaterals
Theorem: A quadrilateral is inscribed in a circle if and only if the opposite angles are supplementary.