Download Converse of the Triangle Proportionality Theorem

Primary Type: Formative Assessment
Status: Published
This is a resource from CPALMS (www.cpalms.org) where all educators go for bright ideas!
Resource ID#: 72160
Converse of the Triangle Proportionality Theorem
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
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.
page 1 of 3 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
in order to show
that are congruent which allows for the conclusion that
because of SAS Similarity Theorem. (2) Identify a pair of corresponding angles
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
to conclude that
Combines two algebraic steps when reasoning from
to
by the SAS Similarity Theorem.
.
Questions Eliciting Thinking
Did you justify all of your statements? Is there any additional explanation or reasoning you can provide?
page 2 of 3 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
and
because of the Reflexive Property of Equality. Then the student reasons that if
. If
then
then there exists some value r such that
. But since AB = AF + FB (by the Segment Addition Postulate), then
. By the same reasoning, AC = (r + 1)AG which indicates that
Similarity Theorem. Because Corresponding Angles of Similar Triangles are congruent,
. It then follows that
so that
because of the SAS
by the Converse of the Corresponding Angles
Theorem.
Questions Eliciting Thinking
Could you have reasoned from
to
in another way? Explain why or why not?
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
O. If
and
, then
and
. Let P and Q be two points so that
does not contain
.
Consider implementing MFAS task Triangle Proportionality Theorem (G-SRT.2.4) if not previously used.
ACCOMMODATIONS & RECOMMENDATIONS
Special Materials Needed:
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
Access Privileges: Public
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.
page 3 of 3