The Size of Inscribed Angles and Central Angles

1. Inscribed Angle
   In circle , for a point on the circumference not lying on arc , is called the inscribed angle subtended by arc , and arc is the arc subtended by inscribed angle .
   
   
   There are infinitely many inscribed angles subtended by arc depending on the position of point .
   
   
2. Relationship Between Inscribed and Central Angles
   In a circle, the size of all inscribed angles subtended by the same arc is equal, and it is half the size of the central angle subtended by the same arc.

Properties of Inscribed Angles

1. The size of inscribed angles subtended by the same arc is equal.
   
   
2. The size of an inscribed angle subtended by a semicircle is .
   is the diameter of circle , so .
   Note: If the inscribed angle subtended by is , then is a semicircle.
   
   

The Size of Inscribed Angles and the Length of Arcs

In a circle:

1. The size of inscribed angles subtended by arcs of equal length is the same.
   implies .
   
   
2. The length of arcs subtended by inscribed angles of the same size is equal.
   implies .
   
   
3. The length of an arc is directly proportional to the size of the inscribed angle subtended by it.
4. Since the length of an arc is directly proportional to the size of the central angle subtended by it, the length of an arc is also directly proportional to the size of the inscribed angle subtended by it.
5. The relationship between the size of an inscribed angle and the length of the corresponding arc holds for congruent circles as well.
6. The size of an inscribed angle and the length of the corresponding chord are not directly proportional.