Getting Started

The student is unable to draw or distinguish between an inscribed angle and a central angle of a circle.

The student does not correctly relate the location of the vertex for each type of angle with the parts of the circle.

The student:

• Draws a central angle but does not identify it as such (and uses incorrect terminology).
  
  
  
  
• Indicates that a central and an inscribed angle are the same angle.
  
  
  
  
• Draws the inscribed angle incorrectly (places the vertex in the circle).
  
  

What is a central angle? Did you draw one?

What is an inscribed angle? Did you draw one?

What does it mean for an angle to intercept an arc?

What arcs do your angles intercept?

Review terminology related to angles of a circle (e.g., central angle, inscribed angle, intercepted arc, semicircle, diameter, and center) and the definitions and theorems that describe angle measures in relation to intercepted arcs. Illustrate the differences between a central and an inscribed angle by emphasizing the location of their vertices with respect to the circle (e.g., the vertex of a central angle is the center of a circle and the vertex of an inscribed angle is a point on the circle).

Consider using one of the following sites which enable the student to explore the relationships between angle measure and arc measure:

• Math Open Reference: http://www.mathopenref.com/.
• Direct link to explore the Central Angle: http://www.mathopenref.com/arcangle.html. Select the box marked 'Show central angle measure' to visualize how the central angle measure changes when the arc is adjusted.
• Direct link to explore the Inscribed Angle: http://www.mathopenref.com/circleinscribed.html. The student is able to move the vertex of the inscribed angle around the circle (while the intercepted arc remains fixed) and to observe how the measure of the angle remains the same regardless of its location.
• Direct link to explore the relationship between an inscribed and a central angle that intercept the same arc: http://www.mathopenref.com/arccentralangletheorem.html.

If needed, provide instruction on using correct notation when naming angles and arcs. Address the differences in naming minor and major arcs of a circle.

Making Progress

The student is unable to correctly describe the relationship between the measures of a central and inscribed angle that intercept the same arc.

The student correctly draws and identifies a central angle and an inscribed angle and may describe the location of the vertex for each angle type. However, the student either is unable to describe the relationship between their measures or describes the relationship incorrectly. For example, the student says the two angles have the same measure.

What do you know about the measure of a central angle in relation to its intercepted arc?

Does the inscribed angle that you drew look like it has the same measure as the central angle?

If the vertex of the inscribed angle is labeled B and the measure of arc PQ = , then what are the measures of  and ? What do you think, in general, is the relationship between the measures of an inscribed angle and a central angle that intercept the same arc?

Review the relationship between central angles and their intercepted arcs and inscribed angles and their intercepted arcs. Using the student’s illustration, suggest a measure for one of the two angles or the intercepted arc and ask the student to find the other two measures. Then ask the student to describe the relationship between the measure of a central angle and an inscribed angle that intercept the same arc.

Provide the student with a variety of practice problems in which a central angle and an inscribed angle intercept the same arc. In each problem, give the student only the central angle measure, the inscribed angle measure, or the intercepted arc measure and ask the student to find the other two measures. Include a problem in which the measure of the intercepted arc is given as , and the student is asked to write expressions representing the measures of the central angle and the inscribed angle. Then ask the student to verbally describe the relationship between the measures of a central angle and an inscribed angle that intercept the same arc.

If needed, provide instruction on using correct notation when naming angles and arcs. Address the differences in naming minor and major arcs of a circle.

Consider implementing other MFAS tasks for G-C.1.2.

Got It

The student provides complete and correct responses to all components of the task.

The student correctly draws a central angle and an inscribed angle that intercept . The student explains that the measure of the central angle is the same as the measure of the intercepted arc, but the measure of the inscribed angle is half the measure of the intercepted arc. Consequently, the measure of an inscribed angle is half the measure of a central angle when both intercept the same arc.

Suppose the vertex of an inscribed angle is on the circle between points P and Q (on ) and its sides contain points P and Q. How is the measure of this inscribed angle related to the measure of the inscribed angle you drew?

Ask the student to prove that the measure of an inscribed angle is half the measure of its intercepted arc.

Consider implementing other MFAS tasks for G-C.1.2.