Getting Started 
Misconception/Error The student is unable to correctly identify central and inscribed angles and determine their measures. 
Examples of Student Work at this Level The student:
 Is only able to correctly identify the central angle.
 Correctly finds some angle measures but is unable to identify them or describe relationships among their measures.
 Incorrectly labels the three angles as either inscribed or central.
 Attempts to classify the angles as acute or obtuse.

Questions Eliciting Thinking Can you identify the angle types with respect to the circle?
Can you identify a central angle in the diagram?
Do you know how to find the measure ofÂ ?
Can you identify an inscribed angle in the diagram?
Do you know how to find the measure ofÂ ? 
Instructional Implications Review the definitions of central, inscribed, and circumscribed angles and the relationships between the measure of each angle type and the measure of its intercepted arc(s). Guide the student to focus on the location of the vertex of the angle when identifying its type (e.g., the vertex of a central angle is the center of the circle, the vertex of an inscribed angle is on the circle, and the vertex of a circumscribed angle is in the exterior of the circle).
Consider using one of the following sites which enable the student to explore the relationships between angle measure and arc measure:
Provide a variety of problems in which the student must find the measure of a central angle and an inscribed angle given information about the measures of intercepted arcs.
If needed, provide instruction on using correct notation when naming angles, arcs, and referring to their measures. Address the differences in naming minor and major arcs of a circle. Also address the difference between naming an angle (e.g.,Â ) and referring to the angleâ€™s measure (e.g., m). 
Moving Forward 
Misconception/Error The student is unable to correctly identify the circumscribed angle and determine its measure. 
Examples of Student Work at this Level The student correctly names both the central and inscribed angles and determines their measures. The student is unable to name and correctly find the measure of the circumscribed angle.

Questions Eliciting Thinking What do you know about the measure ofÂ ?
What kind of lines are Â andÂ ?
What arcs doesÂ Â intercept? 
Instructional Implications Review the definition of a circumscribed angle and the relationship between its measure and the measures of its intercepted arcs (e.g., the measure of a circumscribed angle is equal to half the difference in the measures of its intercepted arcs). Guide the student to observe that the sides of a circumscribed angle are each tangent to the circle. The points of tangency separate the circle into the two intercepted arcs. Provide a variety of problems in which the student must find the measure of a circumscribed angle or the measures of one or both intercepted arcs given appropriate information.
Review the following theorems:
 The radius of a circle is perpendicular to a tangent line at the point of tangency.
 The sum of the measures of the interior angles of a quadrilateral isÂ .
Guide the student to observe thatÂ Â andÂ Â are opposite angles of quadrilateral ABED. Ask the student to determine the measures ofÂ Â andÂ . Then guide the student to reason that if the sum of the four angles of quadrilateral ABED isÂ Â and one pair of opposite angles sum toÂ Â (since ), then the other pair of opposite angles (e.g.,Â Â andÂ ) sum toÂ Â or are supplementary. Ask the student to use this relationship to determine another way to find the measure ofÂ .
If needed, provide instruction on using correct notation when naming angles, arcs, and referring to their measures. Address the differences in naming minor and major arcs of a circle. Also address the difference between naming an angle (e.g.,Â ) and referring to the angleâ€™s measure (e.g., m).
Consider implementing other MFAS tasks for GC.1.2. 
Almost There 
Misconception/Error The student is unable to describe, in general, the relationships among the angle pairs. 
Examples of Student Work at this Level The student correctly names the central, inscribed, and circumscribed angles and determines their measures. However, the student is unable to correctly describe, in general, the relationship betweenÂ Â andÂ Â and the relationship betweenÂ Â andÂ .

Questions Eliciting Thinking How did you find the measures ofÂ Â andÂ ? How do their measures compare?
What kind of polygon is ABED? What is the sum of the measures of the interior angles of a quadrilateral?
What kind of angle isÂ ? How do you know this? 
Instructional Implications Ask the student to explain how he or she found the measures ofÂ Â andÂ . Then remind the student that these two angles intercept the same arc. Guide the student to reason that if Â and , then .
Review the following theorems:
 The radius of a circle is perpendicular to a tangent line at the point of tangency.
 The sum of the measures of the interior angles of a quadrilateral isÂ .
Guide the student to observe thatÂ Â andÂ Â are opposite angles of quadrilateral ABED. Ask the student to determine the measures ofÂ Â andÂ . Then guide the student to reason that if the sum of the four angles of quadrilateral ABED is Â and one pair of opposite angles sum toÂ Â (sinceÂ ), then the other pair of opposite angles (e.g.,Â Â andÂ ) sum toÂ Â or are supplementary. Ask the student to use this relationship to determine another way to find the measure ofÂ .
If needed, provide instruction on using correct notation when naming angles, arcs, and referring to their measures. Address the differences in naming minor and major arcs of a circle. Also address the difference between naming an angle (e.g.,Â ) and referring to the angleâ€™s measure (e.g., m). 
Got It 
Misconception/Error The student provides complete and correct responses to all components of the task. 
Examples of Student Work at this Level The student correctly identifies each angle type and its measure. The student writes the following statements forÂ ,Â , andÂ :
 Â is a central angle and its measure is the same as the measure of its intercepted arc: .
 Â is an inscribed angle and its measure is half the intercepted arc:
 Â is a circumscribed angle and its measure is 180 minus the measure of the central angle: .
OR
 Â is an exterior angle and its measure is half the difference between the measures of the major intercepted arc, , and the minor intercepted arc,Â : .

Questions Eliciting Thinking Would any of your calculation methods change if the three angles did not intersect the same arc on the circle? Suppose the rays ofÂ Â are secants rather than tangents? What calculation(s) would change, if any? Illustrate your explanation using a specific example.
Suppose the rays ofÂ Â are a combination of one tangent and one secant? Would you need any additional information to determine the measure ofÂ ? Draw a diagram to illustrate the situation and then respond.
Does the Central Angle Theorem apply to the angles in this diagram? Explain why or why not. 
Instructional Implications Ask the student to use the diagram to prove, in general, that central Â and circumscribedÂ Â are supplementary.
Consider implementing other MFAS tasks for GC.1.2. 