Angles in inscribed quadrilaterals

Learning Objectives Define an inscribed quadrilateral and identify its properties. State and prove the Inscribed Quadrilateral Theorem. Calculate the measure of an unknown angle in an inscribed quadrilateral using the property that opposite angles are supplementary. Set up and solve algebraic equations to find angle measures in inscribed quadrilaterals. Determine if a given quadrilateral can be inscribed in a circle based on its angle measures. Apply the properties of inscribed quadrilaterals to solve multi-step geometric problems. Ever noticed how a camera's aperture opens and closes? 📸 The blades form a polygon inside a circle. What's the special relationship between the angles of that shape? This tutorial explores inscribed quadrilaterals, which are four-sided...

TermDefinitionExample Inscribed PolygonA polygon whose vertices all lie on a single circle.A square with each of its four corners touching the circumference of a circle. Inscribed QuadrilateralA four-sided polygon whose four vertices all lie on a circle. It is also called a cyclic quadrilateral.If points A, B, C, and D are on a circle, then quadrilateral ABCD is an inscribed quadrilateral. Inscribed AngleAn angle formed by two chords in a circle that have a common endpoint on the circle.In a circle with points A, B, and C on its circumference, ∠ABC is an inscribed angle. Intercepted ArcThe portion of the circle that lies in the interior of an inscribed angle.For an inscribed angle ∠ABC, the arc AC that is 'inside' the angle is the intercepted arc. Supplementary AnglesTwo angles...

Inscribed Quadrilateral Theorem If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary. For an inscribed quadrilateral ABCD, this means m∠A + m∠C = 180° and m∠B + m∠D = 180°. Use this rule to find a missing angle when you know its opposite angle, or to set up an equation if the angles are expressed with variables. Converse of the Inscribed Quadrilateral Theorem If the opposite angles of a quadrilateral are supplementary, then the quadrilateral can be inscribed in a circle. Use this to determine if a four-sided figure is 'cyclic' or can have a circle drawn through all its vertices. If you can show that m∠P + m∠R = 180° and m∠Q + m∠S = 180° for a quadrilateral PQRS, then it is an inscribed quadrilateral.