Central angles

Learning Objectives Define a central angle and identify its components, including the vertex, sides, and intercepted arc. State the relationship between the measure of a central angle and the measure of its intercepted arc. Calculate the measure of a central angle given the degree measure of its intercepted arc. Calculate the degree measure of an intercepted arc given the measure of its central angle. Set up and solve algebraic equations to find unknown angle or arc measures. Apply the concept that the sum of all non-overlapping central angles in a circle is 360 degrees. Use the relationship between congruent central angles and congruent arcs to solve problems. Ever wondered how a pizza is cut into perfectly equal slices or how a clock's hands mark the time? 🍕 The se...

TermDefinitionExample CircleThe set of all points in a plane that are at a fixed distance (the radius) from a fixed point (the center).A circle named Circle P has its center at point P. Central AngleAn angle whose vertex is the center of a circle and whose sides are two radii of the circle.In a circle with center O and points A and B on the circle, the angle ∠AOB is a central angle. ArcAn unbroken part of the circumference of a circle.The portion of the circle from point A to point B is called arc AB. Intercepted ArcThe arc that lies in the interior of a central angle and has endpoints on the sides of the angle.For central angle ∠AOB, the intercepted arc is the arc AB that lies between points A and B. Degree Measure of an ArcThe measure of an arc, in degrees, which is equal to the measure...

Central Angle-Arc Equality m∠AOB = m(arc AB) The measure of a central angle is equal to the measure of its intercepted minor arc. This is the fundamental rule for central angles. Sum of Central Angles Σ(central angles) = 360° The sum of the measures of all non-overlapping central angles that make up a full circle is 360 degrees. This is useful for finding a missing angle when others are known. Congruent Central Angles and Arcs Theorem In the same circle, ∠AOB ≅ ∠COD if and only if arc AB ≅ arc CD. This means that if two central angles are equal in measure, their intercepted arcs are also equal in measure, and vice-versa. This is useful for proofs and solving for unknowns.