Download Triangle Sum Proof

Primary Type: Formative Assessment
Status: Published
This is a resource from CPALMS (www.cpalms.org) where all educators go for bright ideas!
Resource ID#: 60310
Triangle Sum Proof
Students are asked prove that the measures of the interior angles of a triangle sum to 180°.
Subject(s): Mathematics
Grade Level(s): 9, 10, 11, 12
Intended Audience: Educators
Freely Available: Yes
Keywords: MFAS, proof, triangle sum theorem, triangle interior angle sum, alternate interior angles
Resource Collection: MFAS Formative Assessments
ATTACHMENTS
MFAS_TriangleSumProof_Worksheet.docx
MFAS_TriangleSumProof_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 problem on the Triangle Sum Proof worksheet.
2. The teacher asks follow-up questions, as needed.
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 any further.
page 1 of 5 Says that three angles of any triangle sum to 180° but provides no justification or reasoning.
The student’s proof does not logically flow and may include unnecessary steps and errors.
Questions Eliciting Thinking
Did you think through a plan for your proof before you started? Did you consider what you already know that might help you to prove these angles sum to 180°?
Why do you think
was added to the diagram? Can you conclude anything from this?
Are there some things you have learned about parallel lines that can help you? If two parallel lines are intersected by a transversal, what do you know about the angles
formed?
Instructional Implications
Review the meaning of the following: adjacent, straight, supplementary, and alternate interior in the context of pairs of angles. Encourage the student to lengthen and
extend the parallel lines and the two transversals in the diagram. Ask the student to use the diagram on the worksheet to identify examples of each. Review the Alternate
Interior Angle Theorem and guide the student to conclude that the alternate interior pairs are congruent. Guide the student through a proof of the Triangle Sum Theorem
such as the one suggested in the Got It level of the rubric. Prompt the student to supply the justifications of the statements. Then have the student analyze and describe
the strategy used in the proof.
If necessary, review notation for naming angles (e.g., < 1) and describing angle measures (e.g., m < 1) and guide the student to write equations and congruence
statements using the appropriate notation.
Allow the student to explore the sum of interior angles of triangles using computer software or a graphing calculator. Websites such as www.mathopenref.com provide tools
for the student to interact with geometric figures so the student can explore relationships and avoid misconceptions (Specific link:
http://www.mathopenref.com/triangleinternalangles.html).
Provide the student with frequent opportunities to make deductions using a variety of previously encountered definitions and established theorems. For example, provide
diagrams as appropriate and ask the student what can be concluded as a consequence of:
< A and < B are vertical.
< C and < D are corresponding angles (formed by parallel lines intersected by a transversal).
< 1 and < 2 are a linear pair of angles.
is the bisector of < DEF.
Provide the student with additional examples of proofs of statements about triangles and their angles. Ask the student to prove simple statements and provide feedback.
page 2 of 5 Moving Forward
Misconception/Error
The student’s proof reveals some evidence of an overall strategy, but the student fails to establish major conditions leading to the prove statement.
Examples of Student Work at this Level
The student’s proof establishes the congruence of the alternate interior angles but goes no further.
The student’s work states the alternate interior angles are congruent, but then unnecessary steps are included.
Questions Eliciting Thinking
How does knowing the alternate interior angles are congruent help you in this proof?
What do you know by looking at the diagram? Do you see any angle relationships in the diagram other than the ones created by the parallel lines?
Instructional Implications
Provide proof problems for the student in which the statements and reasons are given separately and the student must arrange the steps into a logical order. Allow the
student opportunities to practice this process in multiple proof formats including flow diagrams, two column proofs, and paragraph proofs.
If necessary, review notation for naming angles (e.g., < 1) and describing angle measures (e.g., m < 1) and guide the student to write equations and congruence
statements using the appropriate notation.
Encourage the student to begin the proof process by developing an overall strategy. Provide another statement to be proven and have the student compare strategies
with another student and collaborate on completing the proof.
Consider implementing MFAS task Isosceles Triangle Proof (G-CO.3.10).
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 state the given in the proof.
page 3 of 5 The student fails to give a reason for the final statement of the proof.
The student’s proof contains a mistake in the given statement and fails to name the three angles forming the straight angle (supplementary angles).
Questions Eliciting Thinking
Does your last step prove that the three angles of the triangle sum to 180°?
There is a slight error in your proof, can you find it?
Look over your proof, did you show appropriate reasons for all of your statements?
Instructional Implications
Provide the student with direct feedback on his or her proof. Prompt the student to supply justifications or statements that are missing. Correct any misuse of notation.
If necessary, review notation for naming angles (e.g., < 1) and describing angle measures (e.g., m < 1) and guide the student to write equations and congruence
statements using the appropriate notation.
Provide opportunities for the student to determine the flow of a proof. Give the student each step of a proof written on a separate strip of paper and ask the student to
determine the order of the steps so that there is a logical flow to the proof. Ask the student to then provide the justification for each step.
Consider implementing other MFAS proof tasks, Isosceles Triangle 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 presents a convincing proof that alternate interior angles are congruent.
Questions Eliciting Thinking
Could you accomplish this proof without using the Alternate Interior Angles Theorem? How? What other theorems or postulates could you use?
page 4 of 5 What do you think is the justification for adding
to the diagram? How do we know that there is only one line parallel to
through point B?
Instructional Implications
If necessary, review notation for naming angles (e.g., < 1) and describing angle measures (e.g., m < 1) and guide the student to write equations and congruence
statements using the appropriate notation.
Provide the student opportunities to write proofs using a variety of formats some of which include paragraph proofs, flow diagrams, and two-column proofs.
Consider implementing other MFAS proof tasks, Isosceles Triangle Proof (G-CO.3.10) and Triangle Midsegment Proof (G-CO.3.10).
ACCOMMODATIONS & RECOMMENDATIONS
Special Materials Needed:
Triangle Sum Proof 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.3.10:
Description
Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of
interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent;
the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a
triangle meet at a point.
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