Download Proving Parallelogram Angle Congruence

Primary Type: Formative Assessment
Status: Published
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Resource ID#: 60824
Proving Parallelogram Angle Congruence
Students are asked to prove that opposite angles of a parallelogram are congruent.
Subject(s): Mathematics
Grade Level(s): 9, 10, 11, 12
Intended Audience: Educators
Freely Available: Yes
Keywords: MFAS, proof, parallelogram, opposite angles, congruence
Resource Collection: MFAS Formative Assessments
ATTACHMENTS
MFAS_ProvingParallelogramAngleCongruence_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 proof on the Proving Parallelogram Angle Congruence worksheet.
2. If the student only shows one pair of opposite angles congruent, ask the student, "Do you think the other pair of opposite angles is also congruent? Why or why not?"
3. The teacher asks follow-up questions, as needed.
Note: Remind the students that a theorem cannot be used in its own proof. If necessary, review the theorems that have been proved prior to the introduction of this
theorem so that the student understands which theorems can be used in this proof. Also, proofs may vary from what is shown on this rubric depending on the theorems
proved prior to task implementation.
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:
page 1 of 4 States the given information, but is unable to go any further.
Uses the theorem to be proven in its own proof.
Questions Eliciting Thinking
What do you know about this figure? What are you trying to prove?
Did you think of a plan for your proof before you started?
What if you added the diagonal
to the diagram? Would that help in any way?
Instructional Implications
Provide the student with the statements of a proof of this theorem and ask the student to supply the justifications. Then have the student analyze and describe the overall
strategy used in the proof.
Emphasize that a theorem cannot be used as a justification in its own proof. Encourage the student to first determine what is available to use in a proof of a particular
statement.
Review the ways to prove two triangles congruent (SSS, SAS, ASA, AAS, HL) and what must be established in order to conclude two triangles are congruent when using
each method. Remind the student that once two triangles are proven congruent, all remaining pairs of corresponding parts can be shown to be congruent.
If necessary, review notation for naming sides (e.g.,
) and describing lengths of sides (e.g., W Y ) and guide the student to use the notation appropriately.
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:
Constructs a diagonal of the parallelogram and attempts to prove the two triangles formed are congruent in order to conclude that opposite angles of the parallelogram are
congruent but is unable to show the triangles are congruent.
Questions Eliciting Thinking
What is your general strategy for this proof?
How can you show the two triangles are congruent?
What do you need to show in order to use the ASA (or SSS) congruence theorem? Have you done this in your proof?
Instructional Implications
Review the ways to prove two triangles congruent (SSS, SAS, ASA, AAS, HL) and what must be established in order to conclude two triangles are congruent when using
each method. Provide more opportunities and experiences with proving triangles congruent.
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.
If necessary, review notation for naming sides (e.g.,
) and describing lengths of sides (e.g., W Y ) and guide the student to use the notation appropriately.
Almost There
Misconception/Error
page 2 of 4 The student fails to establish a condition that is necessary for a later statement.
Examples of Student Work at this Level
The student fails to establish:
The congruence of one pair of angles or sides necessary to use the congruence theorem cited.
That opposite sides are parallel before stating that alternate interior angles are congruent.
The uniqueness of
(or
)
Questions Eliciting Thinking
How did you know you could add the diagonal to your diagram? How do you know this segment is unique?
How can you show that the two triangles are congruent?
What must be true for alternate interior angles to be congruent? Did you state that in your proof?
I see that you stated these triangles are congruent. Can you show me all of the steps needed to apply the theorem you used? Did you include all of them in your proof?
Instructional Implications
Review how to address and justify adding a diagonal to a diagram.
Using a colored pencil or highlighter, encourage the student to mark the statements which support the congruence theorem chosen. Remind the student that each letter
of the theorem name represents a pair of parts that must be shown to be congruent (e.g., if using ASA to prove the triangles congruent, the proof must include showing
the congruence of two pairs of corresponding angles and their included sides and a reason or justification must be provided for each).
If necessary, review notation for naming sides (e.g.,
) and describing lengths of sides (e.g., W Y ) and guide the student to use the notation appropriately.
Consider using MFAS tasks Proving Parallelogram Side Congruence (G-CO.3.11), Proving Parallelogram Diagonals Bisect (G-CO.3.11), Proving a Rectangle Is a Parallelogram (GCO.3.11) or Proving Congruent Diagonals (G-CO.3.11) if not previously used.
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 provides a complete proof with appropriate justification such as:
WXYZ is a parallelogram so by definition
and
and
can be constructed and is unique since through any two distinct points passes a unique line.
because they are alternate interior angles. The Reflexive Property states that
the ASA congruence theorem. Therefore,
WXYZ is a parallelogram so
.
and
. Then,
because of
because corresponding parts of congruent triangles are congruent (or by definition of congruent triangles).
because opposite sides of a parallelogram are congruent.
two distinct points passes a unique line. The Reflexive Property states that
. Then,
can be constructed and is unique since through any
because of the SSS congruence theorem. Therefore,
page 3 of 4 because corresponding parts of congruent triangles are congruent (or by definition of congruent triangles).
WXYZ is a parallelogram so by definition
and
. Then,
can be constructed and is unique since through two distinct points passes a unique line.
because they are alternate interior angles. Using the Angle Addition Postulate we can say that
. Using substitution, we can say that
and that
, so therefore, the opposite angles of a parallelogram are congruent.
Questions Eliciting Thinking
Is there another method you could have used to prove the opposite angles congruent?
You only showed one pair of opposite angles are congruent. How can you show that the other pair is also congruent?
Instructional Implications
Challenge the student to prove other statements about parallelograms, squares, rectangles, and rhombi. Provide the student opportunities to write proofs using a variety of
formats some of which include narrative paragraphs, flow diagrams, and two-column format.
Consider using MFAS tasks Proving Parallelogram Side Congruence (G-CO.3.11), Proving Parallelogram Diagonals Bisect (G-CO.3.11), Proving A Rectangle Is a Parallelogram (GCO.3.11) or Proving Congruent Diagonals (G-CO.3.11) if not previously used.
ACCOMMODATIONS & RECOMMENDATIONS
Special Materials Needed:
Proving Parallelogram Angle 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.3.11:
Description
Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include:
opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and
conversely, rectangles are parallelograms with congruent diagonals.
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