Download Isosceles Triangle Proof

Isosceles Triangle Proof
Resource ID#: 56830
Primary Type: Formative Assessment
This document was generated on CPALMS - www.cpalms.org
Students are asked to prove that the base angles of an isosceles triangle are congruent.
Subject(s): Mathematics
Grade Level(s): 9, 10, 11, 12
Intended Audience: Educators
Freely Available: Yes
Keywords: MFAS, prove, isosceles triangle, congruent base angles
Instructional Component Type(s): Formative Assessment
Resource Collection: MFAS Formative Assessments
ATTACHMENTS
MFAS_IsoscelesTriangleProof_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 problem on the Isosceles Triangle 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 and one or two more statements that fail to establish the
congruence of the base angles.
The student states the base angles must be congruent since the sides opposite them are
congruent (restatement of the isosceles triangle theorem), but no other justification is given.
The student"s proof may include one or more irrelevant or incorrect statements.
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 are congruent?
What if you drew a line segment from point B to the midpoint of
formed are congruent, can you conclude that
?
? If the two triangles
How can you prove the two angles are congruent in a way other than stating the isosceles
triangle theorem?
Instructional Implications
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:




Point M is the midpoint of .
< A and < B are supplementary and m < A = d.
.
is the bisector of < E.
Provide the student with the statements of a proof and ask the student to supply the
justifications. Then have the student analyze and describe the overall strategy used in the
proof.
To prove that the base angles of an isosceles triangle are congruent, guide the student to first
describe an overall strategy [e.g., (1) Add an auxiliary line so that two triangles are formed, (2)
Show the two triangles are congruent, and (3) Conclude the base angles are congruent by
definition of congruent triangles]. Then assist the student in filling in the details of the proof
including justifications. Encourage the student to always begin the proof process by
developing an overall strategy. Assist the student by providing feedback on the strategy.
Emphasize that a theorem cannot be used as a justification in its own proof. Encourage the
student to first question what is available to use in a proof of a particular statement.
Review the ways to prove two triangles congruent (SSS, SAS, ASA, AAS, and 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 concluded to be congruent.
Moving Forward
Misconception/Error
The student’s proof shows 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 attempts to prove the two triangles formed by drawing an altitude, angle bisector,
or median from vertex B are congruent (in order to conclude that
) but fails to do so.
Questions Eliciting Thinking
How do you know you are justified in adding this segment to the diagram?
What do you know as a consequence of this segment being an altitude (or angle bisector or
median)? What can you conclude from this? What properties does it have?
What do you need to show in order to use the SSS (or other relevant) congruence theorem?
Have you done that in your proof?
Suppose you show the triangles are congruent. What will allow you to conclude that these
angles are then congruent?
Instructional Implications
Review the triangle congruence theorems and provide more opportunities and experiences
with proving triangles congruent.
Consider using the NCTM lesson Pieces of Proof
(http://illuminations.nctm.org/Lesson.aspx?id=2561) in which the statements and reasons of a
proof are given separately and the student must rearrange the steps in a logical order.
Encourage the student to use multiple proof formats including flow diagrams, two-column,
and paragraph proofs. 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
another statement to be proven and have the student compare strategies with another student
and to collaborate on completing the proof.
Almost There
Misconception/Error
The student’s proof shows evidence of an overall strategy, but the student fails to establish a
condition that is necessary for a later statement in the proof.
Examples of Student Work at this Level
The student fails to first establish the existence of the midpoint M of
.
and the uniqueness of
The student fails to first establish that the bisector of < B intersects
at some point D.
The student fails to establish the congruence of one pair of angles or sides necessary to use the
congruence theorem cited.
Questions Eliciting Thinking
How did you know you could add this segment to your diagram? How do you know this
segment is unique?
How do you know that this angle bisector will intersect the opposite side of the triangle?
I see you stated these triangles are congruent. Can you show me all of the steps needed to use
the theorem you used? Did you include all of them in your proof?
Instructional Implications
Review how to address and justify adding a point, such as a midpoint, or an auxiliary line 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 SSS to prove
the triangles congruent, the proof must include showing three pairs of corresponding sides are
congruent (and a reason or justification must be provided for each)].
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 devises a complete and correct proof in which he or she (1) Adds an auxiliary line
so that two triangles are formed, (2) Shows the two triangles are congruent, and (3) Concludes
the base angles are congruent by definition of congruent triangles. For example, the student
establishes the existence of point M, the midpoint of
, and the uniqueness of
two distinct points passes a unique line"). The student states that
("Through
(as given),
(by the Reflexive Property), and
(by definition of a midpoint). The student
then states that
by the SSS Congruence theorem. The student concludes that
by the definition of congruent triangles.
Questions Eliciting Thinking
Can you think of another way to prove that the base angles of an isosceles triangle are
congruent? How many different ways could you complete this proof?
Instructional Implications
Challenge the student with statements requiring more complex proofs (e.g. given a diagram
that includes overlapping triangles, ask the student to prove a statement that requires first
proving one pair of triangles congruent in order to name a pair of corresponding parts
congruent needed to show a second pair of triangles congruent).
Encourage the student to assist other students in developing and writing proofs.
ACCOMMODATIONS & RECOMMENDATIONS
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Special Materials Needed:
o
Isosceles Triangle 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
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.