Inscribed angles

Learning Objectives Define an inscribed angle, its vertex, its sides, and its intercepted arc. State and apply the Inscribed Angle Theorem to find the measure of an angle or an arc. Prove and apply the theorem that two inscribed angles intercepting the same arc are congruent. Apply the corollary that an angle inscribed in a semicircle is a right angle. State and apply the theorem for inscribed quadrilaterals, proving that opposite angles are supplementary. Solve multi-step problems involving inscribed angles, including those with algebraic expressions. Ever wondered how a satellite dish focuses signals or how a camera's wide-angle lens captures a broad view? 🛰️ The geometry of circles, specifically inscribed angles, holds the key! This tutorial explores inscribed angle...

TermDefinitionExample Inscribed AngleAn angle whose vertex is on a circle and whose sides are chords of the circle.In circle O, if points A, B, and C are on the circle, then ∠ABC is an inscribed angle. Intercepted ArcThe arc that lies in the interior of an inscribed angle and has endpoints on the angle's sides.For inscribed angle ∠ABC, the intercepted arc is arc AC (the part of the circle that is 'inside' the angle). Central AngleAn angle whose vertex is the center of the circle and whose sides are two radii.In circle O, if A and C are on the circle, ∠AOC is the central angle corresponding to arc AC. Its measure is equal to the measure of arc AC. ChordA line segment whose endpoints both lie on the circle.In inscribed angle ∠ABC, the sides AB and BC are both chords of the ci...

The Inscribed Angle Theorem m∠Inscribed = (1/2) * m(Intercepted Arc) The measure of an inscribed angle is exactly half the measure of its intercepted arc. This is the foundational rule for solving most inscribed angle problems. Angles Intercepting the Same Arc If ∠ABC and ∠ADC both intercept arc AC, then ∠ABC ≅ ∠ADC. If two or more inscribed angles in the same circle 'cut out' the same arc, then those angles must be equal in measure. This is a direct consequence of the Inscribed Angle Theorem. Angle in a Semicircle Theorem If an inscribed angle intercepts a semicircle, then the angle is a right angle (90°). This is a special case of the Inscribed Angle Theorem. Since a semicircle is an arc that measures 180°, the inscribed angle that intercepts it must be h...