Getting Started 
Misconception/Error The student is unable to apply the relevant theorems to find the missing angle measures. 
Examples of Student Work at this Level The student is unable to find some or all of the missing angle measures. The student:
 May find the measures of and since each “looks the same” as a given angle but is unable to find the measures of and .
 May not be able to find any of the angle measures.

Questions Eliciting Thinking Do you know what this pair of angles (pointing to a pair of vertical angles) is called? How do you know these angles have the same measure?
Do you know what a straight angle is? What is the measure of a straight angle? What should be true of three angles that combine to form a straight angle? 
Instructional Implications Review the definition of vertical angles and the Vertical Angles Theorem (i.e., vertical angles are congruent). Introduce a proof of the Vertical Angles Theorem and then give the student opportunities to apply this theorem in a variety of problem contexts.
Review the definition of a linear pair and the Linear Pair Postulate. Provide examples of linear pairs of angles. Give the student opportunities to find missing angle measures in diagrams involving linear pairs of angles. 
Moving Forward 
Misconception/Error The student can find the missing angle measures but is unable to adequately justify his or her answers. 
Examples of Student Work at this Level The student correctly finds all missing angle measures. However, when justifying the answers, the student:
 Uses incorrect or unconventional terminology to describe vertical angles and their relationship, such as an angle is “equal to the other side,” the “opposite angle is 25,” the “angle across has the same degree measure,” or “it is a copy of
 Says angles are congruent without providing a justification.
The student may show how the measure of or was computed but offers no justification for the work.

Questions Eliciting Thinking Do you know the mathematical term used to describe and the 25° angle? How do you know these angles are congruent? What theorem supports this statement?
If you combine , the 104° angle, and , what kind of angle is formed? What do you know about this kind of angle?
What does it mean to justify your work? What theorems have you used in finding these angle measures? 
Instructional Implications Review the terms that apply to the angles and the angle relationships in the diagram and their definitions (e.g., vertical angles, linear pair of angles, and supplementary angles). Review postulates and theorems that will be needed in the justifications (e.g., the Vertical Angles Theorem, Angle Addition Postulate, and Linear Pairs Postulate). Explain that when justifying mathematical work, the student should cite relevant definitions, postulates, and theorems that support computational work. For example, model explaining that and the 25° angle are vertical and are therefore, congruent by the Vertical Angles Theorem (i.e., vertical angles are congruent). Guide the student to be explicit about why, for example, and the 104° angle have measures that sum to 129° (i.e., Angle Addition Postulate). Assist the student in observing that this 129° angle and form a linear pair and consequently, have measures that sum to 180°. Ask the student to provide a justification for this conclusion (i.e., Linear Pairs Postulate).
Provide additional opportunities for the student to find missing angle measures using similar diagrams and to justify his or her work. 
Almost There 
Misconception/Error The student is unable to cite relevant definitions, postulates, or theorems that support some aspect of his or her work. 
Examples of Student Work at this Level The student correctly calculates each angle measure and provides a justification for each calculation. However, the student is unable to cite a relevant definition, postulate, or theorem that supports some aspect of his or her work. For example, the student:
 States that and the 25° angle are vertical so must be congruent. However, when asked how he or she knows that vertical angles are congruent, the student is unable to cite the relevant theorem.
 Can justify why vertical angles are congruent (either on paper or when later asked) by citing the relevant theorem but does not completely justify why .
 States that since these angles are “supplementary.” When reminded that the term supplementary only applies to angle pairs, the student is unable to provide an appropriate justification.

Questions Eliciting Thinking How do you know vertical angles are congruent?
How do you know the and 104° sum to 129°? What postulate supports this conclusion?
What does supplementary mean? Can three angles be supplementary?
What do you know about a linear pair of angles? How do you know that a linear pair of angles is supplementary? 
Instructional Implications Provide feedback to the student concerning any error or omission in his or her justification. Explain that a complete justification includes any relevant definitions, postulates, or theorems that support conclusions drawn about angle measures or equations written that model angle relationships. Provide additional opportunities for the student to find missing angle measures using similar diagrams and to justify his or her work. 
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 the 25° angle are vertical and that and the 104° angle are vertical. The student determines that and and justifies this conclusion by citing the Vertical Angles Theorem. The student then determines the measure of either or by observing that, for example, and the 104° angle combine to form a 129° angle (by the Angle Addition Postulate). This angle and form a linear pair. The student states that if two angles form a linear pair, then they are supplementary (by the Linear Pairs Postulate). Consequently, their measures sum to 180° (by definition of supplementary angles). The student writes an equation such as and solves it to determine that the . The student reasons that since and are vertical, is also 51°. 
Questions Eliciting Thinking What is the sum of the measures of all of the angles in this diagram? How might you justify your answer?
You said that if two angles are vertical, then they are congruent. Is it true that if two angles are congruent, then they must be vertical?
Can you think of a realworld application of the Vertical Angles Theorem? 
Instructional Implications Ask the student to prove the Vertical Angles Theorem (i.e., vertical angles are congruent). Consider implementing MFAS task Proving the Vertical Angles Theorem (GCO.3.9).
Introduce special angle pairs formed by parallel lines intersected by a transversal (i.e., alternate interior angles, corresponding angles, and sameside interior angles). Ask the student to conjecture the relationship between alternate interior angles. Then introduce the student to a rigid motion proof of this relationship (i.e., when two parallel lines are intersected by a transversal, alternate interior angles are congruent). Consider implementing MFAS task Proving the Alternate Interior Angles Theorem (GCO.3.9). 