Primary Type: Formative Assessment
Status: Published
This is a resource from CPALMS (www.cpalms.org) where all educators go for bright ideas!
Resource ID#: 62205
Justifying SAS Congruence
Students are asked to use rigid motion to explain why the SAS pattern of congruence ensures triangle congruence.
Subject(s): Mathematics
Grade Level(s): 9, 10, 11, 12
Intended Audience: Educators
Freely Available: Yes
Keywords: MFAS, SAS, side-angle-side, congruence, rigid motions, triangles, transformations
Resource Collection: MFAS Formative Assessments
ATTACHMENTS
MFAS_JustifyingSASCongruence_Worksheet.docx
FORMATIVE ASSESSMENT TASK
Instructions for Implementing the Task
This task can be implemented individually, with small groups, or with the whole class.
1. The teacher asks the student to complete the problem on the Justifying SAS Congruence worksheet.
2. The teacher asks follow-up questions, as needed.
TASK RUBRIC
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.
page 1 of 4 Provides a vague or incomplete argument.
Says the triangles are congruent because of the SAS congruence Theorem.
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 about congruent angles or sides) 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 sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle (SAS). Make clear how
each of the assumptions is used in the explanation and that a statement cannot be used in its own proof.
Explore other congruence postulates (e.g., ASA, 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 one or two vertices of one triangle onto corresponding vertices of
the other but fails to justify that the remaining vertices coincide. For example, the student describes a translation that results in point B coinciding with point E followed by a
rotation that results in
aligning with
. However, the student then restates the given and concludes the triangles are congruent rather than showing how the
remaining vertices 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 will map one triangle onto the other?
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
page 2 of 4 by explaining why two triangles must be congruent when two sides and the included angle of one triangle are congruent to two sides and the included angle of another
triangle (SAS). Make clear how each of the assumptions is used in the explanation and that a statement cannot be used in its own proof.
Explore other congruence postulates (e.g., ASA, 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 rotating
but does not indicate a center or direction of rotation.
States that points A and D will coincide without using the assumption that
.
Does not appeal to any properties of rigid motion such as the preservation of angle measure.
Questions Eliciting Thinking
About what point should
be rotated? How far should it be rotated?
I understand from your explanation why points A and D will lie on the same line, but how do you know they will coincide?
How do you know
must align with
? Can you explain this in more detail?
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., ASA, 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:
1. Translate point B to point E according to
2. Rotate
so that vertex B coincides with vertex E (B = E).
clockwise about point E until
aligns with
3. Since rigid motion preserves length and
4. Since rigid motion preserves angle measure and
5. Since
.
, vertex A now coincides with point D (A = D).
,
must align with
.
then BC = EF, so that point C now coincides with point F (C = F).
6. 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
How might you describe the degree of the rotation of
,
and
?
about point B?
How do you know that if the vertices coincide, the triangles must be congruent?
Instructional Implications
Alter the orientation of the two triangles, so a translation followed by both a rotation and a reflection is needed to map
to
. Ask the student to adapt his or
her explanation to this case.
Challenge the student to rework his or her explanation into a more formal proof in which details are provided and notation is used appropriately.
page 3 of 4 Consider implementing one of the following MFAS tasks Justifying SSS Congruence (G-CO.2.8) or Justifying ASA Congruence (G-CO.2.8).
ACCOMMODATIONS & RECOMMENDATIONS
Special Materials Needed:
Justifying SAS Congruence worksheet
SOURCE AND ACCESS INFORMATION
Contributed by: MFAS FCRSTEM
Name of Author/Source: MFAS FCRSTEM
District/Organization of Contributor(s): Okaloosa
Is this Resource freely Available? Yes
Access Privileges: Public
License: CPALMS License - no distribution - non commercial
Related Standards
Name
MAFS.912.G-CO.2.8:
Description
Explain how the criteria for triangle congruence (ASA, SAS, SSS, and Hypotenuse-Leg) follow from the definition of
congruence in terms of rigid motions.
page 4 of 4