Question

what theorem states that an angle inscribed in a circle is half the measure of its intercepted arc? a. tangent secant theorem b. central angle theorem c. chord theorem d. inscribed angle theorem which of the following is not a property of quadrilaterals inscribed in a circle? a. all sides are of equal length. b. each angle is half of its intercepted arc. c. opposite angles are supplementary. d. the sum of the arcs of opposite angles is 360 degrees. if angle p and angle r are opposite angles in an inscribed quadrilateral and angle p = 120 degrees, what is angle r? a. 60 degrees b. 30 degrees c. 240 degrees d. 120 degrees

Explanation:

1. The Inscribed - Angle Theorem states that an angle inscribed in a circle is half the measure of its intercepted arc. The Tangent - Secant Theorem deals with angles formed by tangents and secants, the Central - Angle Theorem states that the central angle is equal to the measure of its intercepted arc, and the Chord Theorem is about properties of chords in a circle.
2. For a cyclic quadrilateral (a quadrilateral inscribed in a circle), opposite angles are supplementary, and the sum of the arcs of opposite angles is 360 degrees. The property that each angle is half of its intercepted arc applies to inscribed angles, not to angles of a cyclic quadrilateral as a whole, and cyclic quadrilaterals do not have all sides of equal length in general.
3. In an inscribed quadrilateral, opposite angles are supplementary. If angle \(P = 120^{\circ}\), and angle \(P\) and angle \(R\) are opposite angles, then \(P+R = 180^{\circ}\), so \(R=180 - 120=60^{\circ}\).

Answer:

1. d. Inscribed Angle Theorem
2. a. All sides are of equal length.
3. a. 60 degrees