Download Prove the AA Similarity Theorem

Primary Type: Formative Assessment
Status: Published
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Resource ID#: 73180
Prove the AA Similarity Theorem
Students will indicate a complete proof of the AA Theorem for triangle similarity.
Subject(s): Mathematics
Grade Level(s): 9, 10, 11, 12
Intended Audience: Educators
Freely Available: Yes
Keywords: MFAS, similar triangles, similarity, AA theorem
Resource Collection: MFAS Formative Assessments
ATTACHMENTS
MFAS_ProveTheAASimilarityTheorem_Worksheet.pdf
MFAS_SimilarityWithAngleAndSide_Worksheet.docx
MFAS_SimilarityWithAngleAndSide_Worksheet.pdf
MFAS_SimilarityWithGivenSides_Worksheet.docx
MFAS_SimilarityWithGivenSides_Worksheet.pdf
MFAS_ProveTheAASimilarityTheorem_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 Prove the AA Similarity Theorem worksheet.
2. The teacher asks follow-up questions, as needed.
Note: The following theorems are referenced by name in the rubric:
The Fundamental Theorem of Similarity – Let D be a dilation with center O and scale factor r > 0. Suppose point O does not lie on
dilation D. Then
is parallel to
. Let
be the image of
under
and P’Q’ = r(PQ). For more information and a proof, see Wu (2013) which can be accessed at https://math.berkeley.edu/~wu/CCSS-
Geometry_1.pdf.
The Corresponding Angles Theorem – Corresponding angles formed by parallel lines and a transversal are congruent.
The Angle Side Angle Congruence Theorem – If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle,
then the triangles are congruent.
page 1 of 4 TASK RUBRIC
Getting Started
Misconception/Error
The student is unable to use the definition of similarity in terms of similarity transformations to prove the triangles are similar.
Examples of Student Work at this Level
The student:
Reasons that since
and
, the triangles are similar by the AA Similarity Theorem.
Writes down the given information and one or two other unrelated statements without completing the proof.
Questions Eliciting Thinking
What does the AA Similarity Theorem say? Can a theorem be used in its own proof?
Do you know what is meant by a similarity transformation?
What is the definition of similarity in terms of similarity transformations? How can two triangles be shown to be similar?
Instructional Implications
Review the definition of similarity in terms of similarity transformations and explain how the definition can be used to show two triangles are similar. Provide opportunities to
the student to show two triangles are similar using the definition. Then clearly state the AA Similarity Theorem and ask the student to identify the assumption and the
conclusion. Be sure the student understands that a theorem cannot be used as a justification in its own proof.
Review each of the following:
The Fundamental Theorem of Similarity,
The Corresponding Angles Theorem, and
The ASA Congruence Theorem.
Next review the overall strategy of the proof and guide the student through its steps prompting the student for justifications of key statements.
If needed, implement the MFAS tasks Describe the AA Similarity Theorem (G­SRT.1.3) to assess the student’s understanding of the theorem and Justifying a Proof of the
AA Similarity Theorem (G­SRT.1.3) to assess the student’s understanding of a proof of this theorem. Eventually, ask the student to repeat this task.
Moving Forward
Misconception/Error
The student is unable to complete the proof.
Examples of Student Work at this Level
The student attempts to show that the triangles are similar using the definition of similarity in terms of similarity transformations. However, the student is unable to produce
a complete and correct proof. The student provides:
A labeled sketch of two triangles and at least one step that could lead to a proof but then is unable to continue.
A few steps that suggest a strategy but omits key steps and details.
Questions Eliciting Thinking
What is your overall strategy for this proof? What do you need to prove now?
What is the definition of similarity in terms of similarity transformations?
If one figure is the result of a dilation of another figure, are the figures similar?
If one figure is the result of a dilation and a congruence of another figure, are the figures similar?
How is a segment and its image related under a dilation? Do you remember the Fundamental Theorem of Similarity?
Instructional Implications
Review each of the following:
The Fundamental Theorem of Similarity,
the Corresponding Angles Theorem, and
the ASA Congruence Theorem.
Next review the overall strategy of the proof and guide the student through its steps prompting the student for justifications of key statements.
If needed, implement the MFAS task Justifying a Proof of the AA Similarity Theorem (G­SRT.1.3) to assess the student’s understanding of a proof of this theorem.
Eventually, ask the student to repeat this task.
Almost There
Misconception/Error
page 2 of 4 The student provides a correct response but with insufficient reasoning or imprecise language.
Examples of Student Work at this Level
The student shows the triangles are similar using the definition of similarity in terms of similarity transformations. However, the student’s proof contains some errors or
omissions. For example, the student omits or provides incorrect justification for one or two statements or omits a key step of the proof.
Questions Eliciting Thinking
What is the relationship between these two points and their images?
How is a segment and its image related under a dilation?
What is the definition of similarity in terms of similarity transformations?
If one figure is the result of a dilation of another figure, are the figures similar?
If one figure is the result of a dilation and a congruence of another figure, are the figures similar?
Instructional Implications
Provide feedback to the student concerning any errors or omissions and allow the student to revise his or her proof. Consider implementing the MFAS task Justifying a Proof
of the AA Similarity Theorem (G-SRT.1.3) for further experience with the theorem and its proof. Eventually, ask the student to repeat this task.
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 shows the triangles are similar using the definition of similarity in terms of similarity transformations. The student may provide a proof such as the following:
Let
point on
1.
2.
be the point on
so that
such that D(C) =
is parallel to
(which is also equal to
) by D, and let
be the
. Then:
by the Fundamental Theorem of Similarity.
by the Corresponding Angles Theorem.
3.
4.
. Denote the dilation with center A and scale factor r =
by the ASA Congruence Theorem. Call this congruence G.
by the definition of similarity (
5. Since dilation D maps to
to
is the result of a dilation of
and congruence G maps
).
, then
by the definition of similarity.
Questions Eliciting Thinking
Are there other ways to prove the theorem?
Does the AA Similarity Theorem apply to rectangles? To rhombuses?
Instructional Implications
Ask the student to adapt his or her proof to prove the related theorems on either the Similarity With Given Sides or Similarity With Angle and Side worksheets which are
included in the attachments.
ACCOMMODATIONS & RECOMMENDATIONS
Special Materials Needed:
page 3 of 4 Prove the AA Similarity Theorem worksheet
Similarity with Given Sides worksheet (optional)
Similarity with Angle and Side worksheet (optional)
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-SRT.1.3:
Description
Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.
page 4 of 4