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) 
Construct segment AB & segment AC. 
Measure BAC
2) 
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)
Place point D on the circle. Open the Style menu. Change the from dashed to solid line. Construct segment DB & segment DC Measure CDB.
Construct segment AB & segment AC. Measure BAC
2) 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) Place point D on the circle. Open the Style menu. Change the from dashed to solid line. Construct segment DB & segment DC Measure 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.