Arcs & Angles

An angle with its vertex at the center of a circle is called a central angle. An angle whose sides are chords of a circle and whose vertex is on the circle is called an inscribed angle. In this activity you'll investigate relationships among central angles, inscribed angles, and the arcs they intercept.
Points B & C divide Circle A into two arcs. The shorter arc is called a minor arc and the larger one is called a major arc. A minor arc is named after its endpoints. You can drag points B & C to change the measure of the arc. (Because point B was used to construct the circle; moving it will also change the size of the circle.) 1) Toolbar Image Construct segment AB & segment AC. Toolbar Image Measure BAC 2) Toolbar Image Drag point C around the circle and observe the measures. Pay attention to the differences when the arc is a minor arc & when it is a major arc. Q1 Write a conjecture about the measure of the central angle & the measure of the minor arc it intercepts. Q2 Write a conjecture about the measure of the central angle and the measure of the major arc it intercepts. 3)Toolbar Image Place point D on the circle. Open the Style menu. Toolbar ImageChange the from dashed to solid line. Construct segment DB & segment DC Toolbar ImageMeasure CDB.
4) Drag point C and observe the measures of the arc angle & CDB. Q3 Write a conjecture about the measure of an inscribed angle & the arc it intercepts. 5) Drag point D (without crossing points B or C), and observe the measure of CDB. Q4 Write a conjecture about all inscribed angles and their intercepted arcs. 6) Drag point C so that the measure of the central angle is 180 Q5 Write a conjecture about angles inscribed in a semicircle.