Getting Started 
Misconception/Error The student’s proof shows no evidence of an overall strategy or a logical flow. 
Examples of Student Work at this Level The student:
 Draws the diagonals of the quadrilaterals and asserts the congruence of chords of the circle.
 Restates the conclusion of the theorem (e.g., if a quadrilateral is inscribed in a circle then its opposite angles are supplementary).
 Writes some correct statements about the measures of inscribed angles and arcs, but restates the conclusion of the theorem.

Questions Eliciting Thinking Why are the opposite angles of an inscribed quadrilateral supplementary?
What does supplementary mean?
Can segments be supplementary?
How will the segments and radii help you show that angles are supplementary? 
Instructional Implications Review terminology related to angles of a circle (e.g., central angle, inscribed angle, intercepted arc, and center) and the definitions and theorems that describe angle measures in relation to intercepted arcs. Guide the student through the steps of the proof and ask the student to supply the justifications. Then have the student analyze and describe the overall strategy used in the proof. Emphasize that a theorem cannot be used as a justification in its own proof.
If necessary, review notation for naming angles and arcs and describing their measures. Guide the student to write equations and congruence statements using appropriate notation.
Provide the student with another diagram of a quadrilateral inscribed in a circle. Name the vertices of the quadrilateral using letters different from those used in this task. Ask the student to prove that a pair of opposite angles is supplementary. 
Making Progress 
Misconception/Error The student fails to establish a condition that is necessary for a later statement. 
Examples of Student Work at this Level The student’s work shows evidence of an overall strategy, but the student fails to establish a condition that is necessary for a later statement. For example, the student fails to establish one of the following:

Questions Eliciting Thinking How do the angle measures relate to the measures of their intercepted arcs? Should this be stated in your proof?
How do you know that the measures of and sum to ?
What allowed you to conclude that ? 
Instructional Implications Provide the student with feedback concerning any omissions and prompt the student to supply justification or statements that are missing.
If necessary, review notation for naming angles and arcs and describing their measures. Guide the student to write equations and congruence statements using appropriate notation.
Provide the student with another diagram of a quadrilateral inscribed in a circle. Name the vertices of the quadrilateral using letters different from those used in this task. Ask the student to prove that a pair of opposite angles is supplementary.
Consider implementing other MFAS proof tasks, Proving Vertical Angles Congruent (GCO.3.9), Proving Alternate Interior Angles Congruent (GCO.3.9), Isosceles Triangle Proof (GCO.3.10), and Triangle Midsegment Proof (GCO.3.10). 
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 observes that and are inscribed angles of quadrilateral BCDE. So, m = and m = . Since the two named arcs combine to form the entire circle, . Dividing both sides of this equation by two yields = . Finally, by substitution, m + m = , which means and are supplementary.
The student states that the quadrilateral in the second question cannot be inscribed in a circle because opposite angles are not supplementary.

Questions Eliciting Thinking Can you restate the first problem in the form of a theorem? What is the statement of the theorem in ifthen form? (If a quadrilateral is inscribed in a circle, then each pair of opposite angles is supplementary.)
What is the contrapositive of the statement of the theorem? (If a pair of opposite angles of a quadrilateral is not supplementary, then the quadrilateral cannot be inscribed in a circle.)
How is the contrapositive of a statement related to the statement? (They are logically equivalent.)
How does this help you answer the second question?
Is it possible for one pair of opposite angles of a quadrilateral to be supplementary while the other pair is not? 
Instructional Implications If necessary, review notation for naming angles and arcs and describing their measures. Guide the student to write equations and congruence statements using appropriate notation.
Provide the student opportunities to write proofs using a variety of formats some of which include a narrative format, flow diagrams, and the twocolumn format.
Consider implementing other MFAS proof tasks Proving Vertical Angles Congruent (GCO.3.9), Proving Alternate Interior Angles Congruent (GCO.3.9), Isosceles Triangle Proof (GCO.3.10), and Triangle Midsegment Proof (GCO.3.10). 