Getting Started 
Misconception/Error The student does not understand the need to show that the vertices of one triangle coincide with corresponding vertices of the other triangle in order to show the triangles are congruent. 
Examples of Student Work at this Level The student:
 Attempts to describe a sequence of rigid motions that maps one triangle to the other without regard to the assumptions or the need to establish that corresponding vertices coincide.
 Provides a vague or incomplete argument.
 Says the triangles are congruent because of ASA.

Questions Eliciting Thinking Based on the given information, what can you assume is true about and ?
What does the congruence statement indicate about the corresponding sides and angles? Does the sequence of rigid motions you have listed align the vertices of each triangle as indicated in the congruence statement?
In general, how can you show two triangles are congruent using rigid motion? What do you need to show about their vertices?
Can you use a theorem in its own proof? 
Instructional Implications Allow the student to use transparent paper to initially model a sequence of rigid motions that maps to . Then assist the student in describing each rigid motion in adequate detail. Provide the student with several other pairs of congruent triangles and have the student identify the sequence of rigid motions that maps one triangle to the other. Allow the student to use transparent paper as an aid, if needed.
Explain to the student that a convincing explanation of the congruence of two triangles involving rigid motion includes showing how each of the vertices of one triangle must coincide with corresponding vertices of the other triangle. Consequently, a good first step is to map a vertex of one triangle to a vertex of the other triangle (guided by any assumption that a pair of angles is congruent) since this ensures that a first pair of vertices will coincide. The next step is to use any additional assumptions, the properties of rigid motion, and other useful postulates and theorems to show the remaining two pairs of vertices coincide. Model this process by explaining why two triangles must be congruent when two angles and the included side of one triangle are congruent to two angles and the included side of another triangle (ASA). Make clear how each of the assumptions is used in the explanation and that a statement cannot be used in its own proof. Also emphasize the role of the theorem, “If two lines intersect, they intersect in a unique point,” in showing that the third pair of vertices coincide.
Explore other congruence postulates (e.g., SAS, SSS, HL, and AAS) and guide the student to explain, using rigid motion, why each of these patterns of congruence ensures the congruence of triangles. 
Moving Forward 
Misconception/Error The student provides an incomplete explanation. 
Examples of Student Work at this Level The student understands the need to prove that corresponding vertices align and shows how to map two vertices of one triangle onto corresponding vertices of the other but fails to justify that the remaining two vertices coincide. For example, the student describes a translation that results in points B and E coinciding. Next, the student suggests reflecting across and then rotating about point A until aligns with . The student reasons that since , point A now coincides with point D. The student then concludes that since the triangles are congruent. The student fails to show that vertices C and F must now coincide.
The student may also omit some necessary detail(s) in describing rigid motions. 
Questions Eliciting Thinking Can you describe in more detail how this sequence of rigid motions maps to ?
In general, how can you show two triangles are congruent using rigid motion? What do you need to show about their vertices?
You showed how vertex A coincides with vertex D and how vertex B coincides with vertex E. What about the remaining pair of vertices? How do you know they will coincide? 
Instructional Implications Explain to the student that a convincing explanation of the congruence of two triangles involving rigid motion includes showing how each of the vertices of one triangle must coincide with corresponding vertices of the other triangle after a sequence of rigid motions. Consequently, a good first step is to map a vertex of one triangle to a vertex of the other triangle (guided by any assumption that a pair of angles is congruent) since this ensures that a first pair of vertices will coincide. The next step is to use any additional assumptions, the properties of rigid motion, and other useful postulates and theorems to show the remaining two pairs of vertices coincide. Model this process by explaining why two triangles must be congruent when two angles and the included side of one triangle are congruent to two angles and the included side of another triangle (ASA). Make clear how each of the assumptions is used in the explanation. Also emphasize the role of the theorem, “If two lines intersect, they intersect in a unique point,” in showing that the third pair of vertices coincide.
Explore other congruence postulates (e.g., SAS, SSS, HL, and AAS) and guide the student to explain, using rigid motion, why each of these patterns of congruence ensures the congruence of triangles. 
Almost There 
Misconception/Error The student uses rigid motion to develop a convincing argument but leaves out some detail. 
Examples of Student Work at this Level The student describes a sequence of rigid motions that maps the vertices of one triangle to corresponding vertices of the other triangle. However, the student omits some important detail. For example, the student:
 Suggests reflecting but does not indicate a reflection line.
 States that points B and E will coincide without using the assumption that .
 Does not include in his or her description the center and direction of rotation or describes the center incorrectly.
 Does not reference the uniqueness of the point of intersection of two lines when stating that point F must coincide with point C.

Questions Eliciting Thinking Specifically, where does the line of reflection need to lie? Where is the reflected image located?
I understand from your explanation why points B and E will lie on the same line, but how do you know they will coincide?
About what point did you rotate the triangle? At what point did you stop rotating the same triangle? How could you describe the rotation in more detail?
How do you know point F will coincide with point C? What tells you this must be true? 
Instructional Implications Provide feedback to the student and allow the student to revise his or her explanation. Remind the student to be as clear and concise as possible when describing rigid motions, identifying the centers of rotation, the lines of reflection, and vectors that describe translations. If possible, have the student ask a classmate to read his or her explanation to see if it can be followed as written. Remind the student to always use the given assumptions and be sure to include how each vertex of one triangle coincides with the vertices of the other.
Explain to the student the need for a specific description for the location of the line of reflection and the point of rotation. Demonstrate using graph paper or interactive software how moving the line of reflection or the center of rotation alters the location of the image.
Explore other congruence postulates (e.g., SAS, SSS, HL, and AAS) and guide the student to explain, using rigid motion, why each of these patterns of congruence ensures the congruence of triangles. 
Got It 
Misconception/Error The student provides complete and correct responses to all components of the task. 
Examples of Student Work at this Level A sequence of rigid motions is described that maps vertex B to vertex E (or vice versa). Then using the other two assumptions, the student explains how the remaining two vertices of the two triangles coincide providing adequate explanation and justification. For example, the student might explain:
 Translate point A to point D according to so that vertex A coincides with vertex D (A = D).
 Reflect across and then rotate counterclockwise about point A until side aligns with .
 Since rigid motion preserves length and , vertex B now coincides with point E (B E). Since rigid motion preserves angle measure and , must align with .
 Likewise, given that , must align with .
 Since two rays intersect in a unique point, point F, the intersection of and must coincide with point C, the intersection of and (C = F).
 Since vertices A, B, and C coincide with vertices D, E, and F, respectively, must be congruent to .

Questions Eliciting Thinking Where in your explanation did you specifically use the assumptions , and ?
How might you describe the degree of the rotation of about point A?
How do you know that if the vertices coincide, the triangles must be congruent? 
Instructional Implications Challenge the student to rework his or her explanation into a more formal proof in which details are provided and notation is used appropriately.
Consider implementing one of the following MFAS tasks: Justifying SAS Congruence (GCO.2.8) or Justifying SSS Congruence (GCO.2.8). 