Central, inscribed, and circumscribed angles

The  exercise appears under the Geometry Math Mission. This exercise develops the relationships between central, inscribed and circumscribed angles in circles.

Types of Problems
There are four types of problems in this exercise:


 * 1) Find the angle using tangent to a radius: This problem provides a diagram that has a tangent that hits a circle with a radius. The student is expected to use the diagram to find the measure of a labeled angle.Ciaca1.png
 * 2) Find the angle using a doubly tangent quadrilateral: This problem provides a diagram with a doubly tangent quadrilateral to a circle. The student is expected to find a missing angle measure.Ciaca3.png
 * 3) Find the angle using a triangle in a semicircle: This problem provides a diagram that has a triangle inscribed in a semicircle. The student is expected to use the diagram to find the measure of a labeled angle.Ciaca2.png
 * 4) Find the length using a tangent to the circle: This problem provides a diagram with a tangent to a circle. The student is expected to use the diagram to find a length based on the information pictured.Ciaca4.png

Strategies
Knowledge of the relationship between inscribed angles, central angles, and the intercepted are encouraged to ensure success on this exercise.
 * 1) On ''Find the angle using tangent to a radius: the tangent to a radius is perpendicular, so it measures ninety degrees. Thus the Pythagorean Theorem holds for the lengths.
 * 2) On Find the angle using a doubly tangent quadrilateral, since there are two tangents to radii, the other two angles in such a quadrilateral are complementary, i.e., they sum to ninety.
 * 3) On Find the angle using a triangle in a semicircle, since the angle not located on the diameter is inscribed, the measure at that point is ninety degrees.

Real-life Applications

 * 1) Angle and length problems of this type are common on standardized tests such as the SAT, ACT, ASVAB and GRE.
 * 2) Circles are foundational to understanding conics, and eventually trigonometry.