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Converse of the Triangle Proportionality
Theorem
Resource ID#: 72160
Primary Type: Formative Assessment
This document was generated on CPALMS - www.cpalms.org
Students are asked to prove that if a line intersecting two sides of a triangle divides those two sides
proportionally, then that line is parallel to the third side.
Subject(s): Mathematics
Grade Level(s): 9, 10, 11, 12
Intended Audience: Educators
Freely Available: Yes
Keywords: MFAS, triangle proportionality theorem, side splitter theorem, proof, converse
Instructional Component Type(s): Formative Assessment
Resource Collection: MFAS Formative Assessments
ATTACHMENTS
MFAS_ConverseOfTheTriangleProportionalityTheorem_Worksheet.docx
MFAS_ConverseOfTheTriangleProportionalityTheorem_Worksheet.pdf
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 proof on the Converse of the Triangle
Proportionality Theorem worksheet.
2. The teacher asks follow-up questions, as needed.
Note: Remind the student that a theorem cannot be used in its own proof. If necessary, review the
theorems that have been proven prior to the introduction of this theorem, so the student understands
which theorems can be used in this proof.
TASK RUBRIC
Getting Started
Misconception/Error
The student’s proof shows no evidence of an overall strategy or logical flow.
Examples of Student Work at this Level
The student:

States the given information but is unable to go further.


States that
because of the Converse of the Triangle Proportionality Theorem.
Writes several statements that may or may not be relevant or make sense.
Questions Eliciting Thinking
What do you know about this figure? What are you being asked to prove?
What type of plan did you develop for your proof before you started?
If two lines are intersected by a transversal, what do you know about the angles formed? Can you
identify any of those angles in the diagram? What would need to be true about the angles in order
for the two lines to be parallel?
What do you know about the corresponding angles of similar triangles? What would we need to
show in order to prove the two triangles are similar?
Instructional Implications
Assist the student in understanding what can be used in a proof of a theorem (i.e., the assumptions
presented in the statement of the theorem as well as definitions, postulates, and other previously
established theorems). Emphasize that a theorem cannot be used as a justification in its own proof.
Assist the student in devising an overall strategy for the proof: (1) Establish that
order to show
because of SAS Similarity Theorem. (2) Identify a pair of
in
corresponding angles that are congruent which allows for the conclusion that
by the
Converse of the Corresponding Angles Theorem. Then review the theorems, postulates and
definitions needed to complete the proof (e.g., SAS Similarity Theorem, Converse of the
Corresponding Angles Postulate, Segment Addition Postulate, Substitution Property of Equality,
Reflexive Property). Guide the student through the statements of the proof and prompt the student
to supply the justifications.
Provide proofs of other theorems in which the statements and reasons are given separately and the
student must rearrange the steps into a logical order (e.g., a proof of the Triangle Midsegment
Theorem). Consider using the NCTM lesson Pieces of Proof
(http://illuminations.nctm.org/Lesson.aspx?id=2561). Allow the student to work with a partner to
complete these exercises.
Encourage the student to begin the proof process by developing an overall strategy. Provide
opportunities for the student to determine the flow of a proof. Provide another statement to be
proven and have the student compare strategies with another student, and collaborate on
completing the proof.
Consider implementing other MFAS proof tasks, Isosceles Triangle Proof (G-CO.3.10), Triangle
Sum Proof (G-CO.3.10), or Triangle Midsegment Proof (G-CO.3.10). If needed, provide the
student with the steps of the proofs but with several of the statements and/or justifications
missing.
Moving Forward
Misconception/Error
The student’s proof shows evidence of an overall strategy but fails to establish major conditions
leading to the prove statement.
Examples of Student Work at this Level
The student recognizes the need to show that
in order to reason that a pair of
corresponding angles is congruent and the lines are parallel but is unable to do so.
Questions Eliciting Thinking
What was your overall plan for this proof?
What methods are there for showing two triangles are similar?
What proportion are you trying to establish?
How can you reason from
to
?
Instructional Implications
Review the theorems, postulates and definitions needed to complete the proof (e.g., SAS
Similarity Theorem, Converse of the Corresponding Angles Postulate, Segment Addition
Postulate, Substitution Property of Equality, Reflexive Property). Review an overall strategy for
the proof and guide the student through the steps of any aspect of his or her proof that was
incomplete. Prompt the student to provide justifications for each step.
Provide another statement to be proven. Encourage the student to begin the process of writing a
proof by developing an overall strategy and have the student compare his or her strategies with the
strategies of another student at the same level. For additional practice, provide other theorems to
be proven in which the statements and reasons are given separately and the student must rearrange
the steps into a logical order.
Consider implementing other MFAS proof tasks, Isosceles Triangle Proof (G-CO.3.10), Triangle
Sum Proof (G-CO.3.10), or Triangle Midsegment Proof (G-CO.3.10).
If necessary, review notation for naming angles (e.g.
instead of
) when more than one
angle shares a common vertex. Also, review that when naming similar triangles, corresponding
parts must be in the same order.
Almost There
Misconception/Error
The student’s proof shows evidence of an overall strategy, but the student fails to establish a
minor condition that is necessary to prove the theorem.
Examples of Student Work at this Level
The student:


Fails to give a reason for one or more statements of the proof.
Does not explicitly indicate that AB = AF + FB and AC = AG + GC by the Segment
Addition Postulate.

Does not include the statement that
Similarity Theorem.

Combines two algebraic steps when reasoning from
to conclude that
by the SAS
to
.
Questions Eliciting Thinking
Did you justify all of your statements? Is there any additional explanation or reasoning you can
provide?
Why are you able to replace AB with AF + FB ? Do the statements you included justify your
reasoning?
I see you stated these triangles are similar. Can you show me all of the steps needed to use this
theorem? Did you include all of them in your proof?
Can you explain how you reasoned from
to
?
Instructional Implications
Provide the student with feedback on his or her proof. If the student omitted a statement, have the
student go through each step of the proof to see if he or she can find the gap in the logical flow of
the proof. Prompt the student to supply justifications or statements that are missing. If needed,
review the theorems, postulates and definitions needed to complete the proof (e.g., SAS Similarity
Theorem, Converse of the Corresponding Angles Postulate, Segment Addition Postulate,
Substitution Property of Equality, Reflexive Property). Encourage the student to correct any
misuse of notation, including misleading abbreviations of justifications. If necessary, review
notation for naming angles (e.g.
instead of
) when more than one angle shares a common
vertex. Also, review that when naming similar triangles, vertices are named in corresponding
order.
Consider implementing MFAS tasks Isosceles Triangle Proof (G-CO.3.10), Triangle Sum Proof
(G-CO.3.10), and Triangle Midsegment Proof (G-CO.3.10).
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 states that
reasons that if
If
then
Postulate), then
indicates that
because of the Reflexive Property of Equality. Then the student
then there exists some value r such that
and
.
. But since AB = AF + FB (by the Segment Addition
. By the same reasoning, AC = (r + 1)AG which
. It then follows that
because of the SAS Similarity
Theorem. Because Corresponding Angles of Similar Triangles are congruent,
that
so
by the Converse of the Corresponding Angles Theorem.
Questions Eliciting Thinking
Could you have reasoned from
not?
to
in another way? Explain why or why
Can you state some additional proportions that must be true?
Instructional Implications
Challenge the student to use the Fundamental Theorem of Similarity to show that
.
Note: The Fundamental Theorem of Similarity is as follows: Let D be a dilation with center O and
scale factor
. Let P and Q be two points so that
and
, then
and
does not contain O. If
.
Consider implementing MFAS task Triangle Proportionality Theorem (G-SRT.2.4) if not
previously used.
ACCOMMODATIONS & RECOMMENDATIONS

Special Materials Needed:
o
Converse of the Triangle Proportionality Theorem 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
License: CPALMS License - no distribution - non commercial
Related Standards
Name
MAFS.912.G-SRT.2.4:
Description
Prove theorems about triangles. Theorems include: a line parallel
to one side of a triangle divides the other two proportionally, and
conversely; the Pythagorean Theorem proved using triangle
similarity.