Download Inscribed Quadrilaterals

Primary Type: Formative Assessment
Status: Published
This is a resource from CPALMS (www.cpalms.org) where all educators go for bright ideas!
Resource ID#: 70974
Inscribed Quadrilaterals
Students are asked to prove that opposite angles of a quadrilateral, inscribed in a circle, are supplementary.
Subject(s): Mathematics
Grade Level(s): 9, 10, 11, 12
Intended Audience: Educators
Freely Available: Yes
Keywords: MFAS, inscribed quadrilaterals, opposite angles, supplementary, circles, proof, inscribed angles,
intercepted arcs
Resource Collection: MFAS Formative Assessments
ATTACHMENTS
MFAS_InscribedQuadrilaterals_Worksheet.docx
MFAS_InscribedQuadrilaterals_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 problems on the Inscribed Quadrilaterals 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 a logical flow.
Examples of Student Work at this Level
The student:
Draws the diagonals of the quadrilaterals and asserts the congruence of chords of the circle.
page 1 of 4 ​
Restates the conclusion of the theorem (e.g., if a quadrilateral is inscribed in a circle then its opposite angles are supplementary).
Writes some correct statements about the measures of inscribed angles and arcs, but restates the conclusion of the theorem.
Questions Eliciting Thinking
Why are the opposite angles of an inscribed quadrilateral supplementary?
What does supplementary mean?
Can segments be supplementary?
How will the segments and radii help you show that angles are supplementary?
Instructional Implications
Review terminology related to angles of a circle (e.g., central angle, inscribed angle, intercepted arc, and center) and the definitions and theorems that describe angle
measures in relation to intercepted arcs. Guide the student through the steps of the proof 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.
If necessary, review notation for naming angles and arcs and describing their measures. Guide the student to write equations and congruence statements using appropriate
notation.
Provide the student with another diagram of a quadrilateral inscribed in a circle. Name the vertices of the quadrilateral using letters different from those used in this task. Ask
the student to prove that a pair of opposite angles is supplementary.
Making Progress
Misconception/Error
The student fails to establish a condition that is necessary for a later statement.
Examples of Student Work at this Level
The student’s work shows evidence of an overall strategy, but the student fails to establish a condition that is necessary for a later statement. For example, the student
fails to establish one of the following:
and
.
page 2 of 4 .
.
Questions Eliciting Thinking
How do the angle measures relate to the measures of their intercepted arcs? Should this be stated in your proof?
How do you know that the measures of
and
sum to
What allowed you to conclude that
?
?
Instructional Implications
Provide the student with feedback concerning any omissions and prompt the student to supply justification or statements that are missing.
If necessary, review notation for naming angles and arcs and describing their measures. Guide the student to write equations and congruence statements using appropriate
notation.
Provide the student with another diagram of a quadrilateral inscribed in a circle. Name the vertices of the quadrilateral using letters different from those used in this task. Ask
the student to prove that a pair of opposite angles is supplementary.
Consider implementing other MFAS proof tasks, Proving Vertical Angles Congruent (G-CO.3.9), Proving Alternate Interior Angles Congruent (G-CO.3.9), 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 observes that
and
are inscribed angles of quadrilateral BCDE. So, m
combine to form the entire circle,
=
, which means
=
. Dividing both sides of this equation by two yields
and
and m
=
=
. Since the two named arcs
. Finally, by substitution, m
+m
are supplementary.
The student states that the quadrilateral in the second question cannot be inscribed in a circle because opposite angles are not supplementary.
Questions Eliciting Thinking
Can you restate the first problem in the form of a theorem? What is the statement of the theorem in if-then form? (If a quadrilateral is inscribed in a circle, then each pair of
opposite angles is supplementary.)
What is the contrapositive of the statement of the theorem? (If a pair of opposite angles of a quadrilateral is not supplementary, then the quadrilateral cannot be inscribed
in a circle.)
How is the contrapositive of a statement related to the statement? (They are logically equivalent.)
How does this help you answer the second question?
Is it possible for one pair of opposite angles of a quadrilateral to be supplementary while the other pair is not?
Instructional Implications
If necessary, review notation for naming angles and arcs and describing their measures. Guide the student to write equations and congruence statements using appropriate
page 3 of 4 notation.
Provide the student opportunities to write proofs using a variety of formats some of which include a narrative format, flow diagrams, and the two-column format.
Consider implementing other MFAS proof tasks Proving Vertical Angles Congruent (G-CO.3.9), Proving Alternate Interior Angles Congruent (G-CO.3.9), Isosceles Triangle
Proof (G-CO.3.10), and Triangle Midsegment Proof (G-CO.3.10).
ACCOMMODATIONS & RECOMMENDATIONS
Special Materials Needed:
Inscribed Quadrilaterals 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-C.1.3:
Description
Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral
inscribed in a circle.
page 4 of 4