Download Prove Rhombus Diagonals Bisect Angles

Primary Type: Formative Assessment
Status: Published
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Resource ID#: 65105
Prove Rhombus Diagonals Bisect Angles
Students are asked to prove a specific diagonal of a rhombus bisects a pair of angles.
Subject(s): Mathematics
Grade Level(s): 9, 10, 11, 12
Intended Audience: Educators
Freely Available: Yes
Keywords: MFAS, diagonals, angles, rhombus, bisect
Resource Collection: MFAS Formative Assessments
ATTACHMENTS
MFAS_ProveRhombusDiagonalsBisectAngles_Worksheet.docx
MFAS_ProveRhombusDiagonalsBisectAngles_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 Prove Rhombus Diagonals Bisect Angles 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 does not use triangle congruence or properties of a parallelogram in the proof. The student:
Draws diagonal
but then assumes, rather than proves, that
and
are bisected by diagonal
.
page 1 of 4 Makes some observations about the rhombus without regard to the statement to be proven.
Determines that
and
are bisected by diagonal
by “definition of an angle bisector.”
Uses the statement to be proven as a justification in its own proof.
Questions Eliciting Thinking
What is it that you are given and what are you trying to prove?
Is diagonal
shown in the diagram? Did you think to draw it? If you draw it, what two geometric figures are formed? How can these figures assist you with the planning
of your proof?
What properties of parallelograms or rhombuses might help you with this proof?
What markings are used to indicate congruent parts on a diagram? Can you mark the diagram to show congruent parts? How can these congruent parts help you design a
plan for this proof?
Did you think of a plan for your proof before you started?
Instructional Implications
Review the properties of a rhombus. Ask the student to draw diagonal
and to consider what can be concluded if the two triangles formed are congruent. Assist the
student in devising an overall strategy for the proof: (1) Draw the diagonal to form two triangles. (2) Show the two triangles are congruent. (3) Use the congruence to
conclude that
=
and
=
and complete the proof. Guide the student through the statements of the proof and prompt the student to
supply the justifications.
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.
If needed, 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.
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.
Provide additional opportunities to construct proofs of statements involving congruent or similar triangles.
Moving Forward
Misconception/Error
The student’s proof shows some evidence of an overall strategy but fails to establish major conditions and/or includes incorrect statements.
Examples of Student Work at this Level
The student is unable to show that
but can use this fact to complete the proof.
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 SAS (or SSS) congruence theorem? Have you done this in your proof?
Have you included statements that are unnecessary?
page 2 of 4 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.
Review an overall strategy for the proof and guide the student through the steps of any part of his or her proof that was incomplete. Prompt the student to provide
justifications for each step.
Provide additional opportunities to construct proofs of statements involving congruent or similar triangles.
Almost There
Misconception/Error
The student’s proof contains a minor error.
Examples of Student Work at this Level
The student:
Makes an error in a statement or justification.
Includes unnecessary statements such as
.
Uses notation or names angles incorrectly.
Questions Eliciting Thinking
There is an error in this statement. Can you compare it to the diagram and find the error?
Did you use the fact that
in your proof?
How is the notation for the name of a side different from the notation for the length of a side? How is the notation for the name of an angle different from the notation for
the measure of an angle?
There are three different angles with point A as a vertex. What did you mean by
?
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. If the student included an unnecessary statement, challenge the student to find the statement and remove it. Prompt the
student to supply justifications or statements that are missing. If necessary, review notation for naming sides, lengths of sides, angles, and angle measures. Also, review that
when naming congruent triangles, vertices are named in corresponding order.
Provide additional opportunities to construct proofs of statements involving congruent or similar triangles.
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 uses the SSS Congruence Theorem or the SAS Congruence Theorem to show that
as a consequence of the congruence. The student then concludes that both
and concludes that
and
are bisected by diagonal
and
.
page 3 of 4 Questions Eliciting Thinking
Can you think of another way to prove this statement?
Is this statement also true of diagonal
?
Is this statement generally true of parallelograms? Why or why not?
Instructional Implications
Challenge the student to determine which of the following statements are true:
The diagonals of a rectangle are congruent.
The diagonals of a parallelogram each bisect a pair of opposite angles.
Opposite angles of a kite are congruent.
The diagonals of a square are perpendicular.
Then ask the student to write a proof of the statements that are true.
ACCOMMODATIONS & RECOMMENDATIONS
Special Materials Needed:
Prove Rhombus Diagonals Bisect Angles 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.2.5:
Description
Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
Remarks/Examples:
Geometry - Fluency Recommendations
Fluency with the triangle congruence and similarity criteria will help students throughout their investigations of
triangles, quadrilaterals, circles, parallelism, and trigonometric ratios. These criteria are necessary tools in many
geometric modeling tasks.
page 4 of 4