Getting Started 
Misconception/Error The student is unable to correctly apply relevant theorems to find the missing angle measures. 
Examples of Student Work at this Level The student understands the need to apply the Triangle Sum Theorem but:
 Applies the theorem incorrectly.
 Subtracts 35 from 180 and then is unable to continue.
 Makes a false assumption such as the triangle is right.

Questions Eliciting Thinking What does the Triangle Sum Theorem say? What do the measures of the angles of a triangle sum to?
What is the significance of ? Did you use this fact in your solution?
Can you explain why you think this triangle is a right triangle? 
Instructional Implications Review the Isosceles Triangle Theorem and the Triangle Sum Theorem. Explain the relevance of these theorems to the given problem. Guide the student to apply each theorem to find the missing angle measure. Then show the student a model solution, such as:
Since , then (by the Isosceles Triangle Theorem). Let . Then 2x + 35 = 180° (by the Triangle Sum Theorem) so that x = 72.5°. Therefore, .
Explain each step of the solution, and establish conventions for justifying work (e.g., citing a supporting definition, postulate, or theorem in parentheses near the statement it justifies).
Review other theorems related to triangles (e.g., The measures of the acute angles of a right triangle sum to 90°, or the measure of each angle of an equilateral triangle is 60°.). Then provide opportunities to find missing angle measures in isosceles, right, and equilateral triangles. Ask the student to justify his or her work.
If needed, assist the student in using notation correctly. Be sure the student understands how to correctly write both equations (e.g., ) and congruence statements (e.g., ). 
Moving Forward 
Misconception/Error The student can find the missing angle measure but is unable to adequately justify his or her answers. 
Examples of Student Work at this Level The student correctly determines that the measure of is 72.5°. However, the student’s justification is incorrect or incomplete. For example, the student:
 Fails to cite the Triangle Sum Theorem.
 Fails to cite the Isosceles Triangle Theorem.
 Describes the computations used without providing justification.

Questions Eliciting Thinking How do you know to subtract 35 from 180?
How do you know and are congruent? What supports this statement?
What does it mean to justify your work? What postulates or theorems have you used in finding this angle measure? 
Instructional Implications Explain that a complete justification includes any relevant definitions, postulates, or theorems that support conclusions drawn about angle measures or equations written to model angle relationships. Provide feedback to the student concerning any error or omission in his or her justification. Assist the student in identifying and citing the relevant theorems.
Encourage the student to write and solve an equation to find the missing angle measure. Show the student a model solution, such as:
Since , then (by the Isosceles Triangle Theorem). Let . Then 2x + 35 = 180° (by the Triangle Sum Theorem) so that x = 72.5°. Therefore, .
Provide additional opportunities for the student to find missing angle measures in triangles and to justify his or her work. 
Almost There 
Misconception/Error The student’s solution contains notational errors, a misuse of terminology, or incorrectly written mathematics. 
Examples of Student Work at this Level The student correctly determines that the measure of is 72.5° and justifies the solution by citing both the Triangle Sum Theorem and the Isosceles Triangle Theorem. However, the student makes notational errors, misuses terminology, or incorrectly writes mathematical work.

Questions Eliciting Thinking What are the two congruent angles of an isosceles triangle called? What is the third angle of an isosceles triangle called?
You made some notational errors when you wrote up your work. Can you find and correct them?
Can you find the error in this equation, ? 
Instructional Implications If needed, review terminology used to describe the parts of an isosceles triangle (i.e., base, legs, base angles, and vertex angle). Provide feedback to the student concerning any notational errors or incorrectly written mathematics, and allow the student to revise his or her solution.
Encourage the student to write and solve an equation to find the missing angle measure. Show the student a model solution, such as:
Since , then (by the Isosceles Triangle Theorem). Let . Then 2x + 35 = 180° (by the Triangle Sum Theorem) so that x = 72.5°. Therefore, .
Provide additional opportunities for the student to find missing angle measures in 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 correctly determines that the measure of is 72.5° and justifies the solution by citing both the Triangle Sum Theorem and the Isosceles Triangle Theorem.

Questions Eliciting Thinking How do you know the triangle is isosceles?
How would you find the measure of if the measure of had been given as ? 
Instructional Implications If the student did not do so, ask the student to write and solve an equation to find the missing angle measure.
If not done previously, ask the student to prove the Triangle Sum Theorem and the Isosceles Triangle Theorem. 