Download Justifying the Exterior Angle of a Triangle Theorem

Primary Type: Formative Assessment
Status: Published
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Resource ID#: 70182
Justifying the Exterior Angle of a Triangle Theorem
Students are asked to apply the Exterior Angle of a Triangle Theorem and provide an informal justification.
Subject(s): Mathematics
Grade Level(s): 8
Intended Audience: Educators
Freely Available: Yes
Keywords: MFAS, angle, interior, exterior, adjacent, non-adjacent, supplementary, informal justification
Resource Collection: MFAS Formative Assessments
ATTACHMENTS
MFAS_JustifyingTheExteriorAngleOfATriangleTheorem_Worksheet.docx
MFAS_JustifyingTheExteriorAngleOfATriangleTheorem_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 Justifying the Exterior Angle of a Triangle Theorem worksheet.
2. The teacher asks follow-up questions, as needed.
TASK RUBRIC
Getting Started
Misconception/Error
The student is unable to draw an exterior angle of a triangle.
Examples of Student Work at this Level
The student:
Creates a new angle at vertex B.
page 1 of 5 Labels the interior angle at vertex B as the exterior angle.
Provides no answer or indicates he or she does not understand.
Questions Eliciting Thinking
What is meant by an interior angle of a triangle? Can you show me an example of an interior angle?
What is an exterior angle of a triangle? Can you draw an example?
If I extend
(demonstrate for the student), which angle is the exterior angle?
Instructional Implications
Review the following terms: interior angle of a triangle, exterior angle of a triangle, nonadjacent (or remote) interior angles. Assist the student in extending
and
identifying an exterior angle at vertex B. Show the student that there are two exterior angles at each vertex formed by extending all sides of the triangle at each vertex.
Guide the student to observe that each pair of exterior angles at the same vertex is vertical and, therefore, congruent and that the interior and exterior angle at each
vertex form a linear pair. Provide additional examples of triangles and ask the student to draw an exterior angle at each vertex.
Guide the student in translating the statement of the Exterior Angle Theorem into a specific statement (e.g., equation) about the given diagram. Provide a few more
examples of triangles with exterior angles drawn and ask the student to write equations that show the relationship between the measure of each exterior angle and its two
nonadjacent interior angles.
Assist the student in developing an informal justification of the Exterior Angle Theorem using tracing paper. Then review previously established theorems about angle
relationships, such as: (1) a linear pair of angles is supplementary and (2) the measures of the interior angles of a triangle sum to 180°. Then guide the student to logically
reason about the relationship among the measures of
,
, and
.
Moving Forward
Misconception/Error
The student is unable to apply the Exterior Angle Theorem to write an equation.
Examples of Student Work at this Level
The student is able to extend
and identify the exterior angle formed as
that describes the measure of
. The student:
. However, the student is unable to apply the Exterior Angle Theorem to write an equation
Writes an equation in which sides are set equal to angles.
page 2 of 5 References an angle or a side but does not write an equation.
Writes an incorrect equation.
Questions Eliciting Thinking
Can you show me the location of each angle in your equation? Is it true that the sum of the measures of the three interior angles is equal to
?
What does the Exterior Angle Theorem state? Does your equation model this?
Show me how you would write the sum of the measures of
and
. According to the theorem, what does the sum of those two nonadjacent angles equal?
Instructional Implications
Review the following terms: interior angle of a triangle, exterior angle of a triangle, nonadjacent (or remote) interior angles and assist the student in understanding the
Exterior Angle Theorem. Guide the student in translating the statement of the theorem into a specific statement (e.g., equation) about the given diagram. Provide a few
more examples of triangles with exterior angles drawn and ask the student to write equations that show the relationship between the measure of each exterior angle and
its two nonadjacent interior angles.
Assist the student in developing an informal justification of the Exterior Angle Theorem using tracing paper. Then review previously established theorems about angle
relationships, such as: (1) a linear pair of angles is supplementary and (2) the measures of the interior angles of a triangle sum to 180°. Then guide the student to logically
reason about the relationship among the measures of
,
, and
.
Review how to name angles and sides and how to refer to their measures and lengths. Provide feedback to the student regarding any misuses of notation and ask the
student to revise his or her responses.
Almost There
Misconception/Error
page 3 of 5 The student is unable to provide a clear justification for the theorem.
Examples of Student Work at this Level
The student:
Provides an example rather than a justification.
Provides a statement, whether true or not, that does not justify the theorem. (e.g., “The angles of a triangle put together equal 180°.”)
Provides insufficient justification.
Questions Eliciting Thinking
What do you know about the relationship between an exterior angle and its adjacent interior angle?
What do you know about the sum of the measures of the interior angles of a triangle?
Instructional Implications
Assist the student in developing an informal justification of the Exterior Angle Theorem using tracing paper. Then review previously established theorems about angle
relationships, such as: (1) a linear pair of angles is supplementary and (2) the measures of the interior angles of a triangle sum to 180°. Then guide the student to logically
reason about the relationship among the measures of
,
, and
.
Review how to name angles and sides, and how to refer to their measures and lengths. Provide feedback to the student regarding any misuses of notation and ask the
student to revise his or her responses.
Provide additional opportunities to justify the relationship between the measures of angles. Assist the student in transitioning from using tracing paper to developing a logical
argument.
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 extends
and identifies the exterior angle formed as
. The student writes the equation
and provides an informal justification. For
example, the student:
Traces
and
, then redraws them so they share a vertex and a side. Then the student uses tracing paper to compare the combined angles to
, demonstrating
that they are congruent.
Explains that
and
are supplementary (or
= 180°) since they form a linear pair. The student also observes that (because the three measures of the interior angles of any triangle sum to 180°). The student then reasons that = 180°
so that
.
page 4 of 5 Questions Eliciting Thinking
What other angle relationships could you identify if you draw a line parallel to
through point C and consider
and
as transversals?
Instructional Implications
If the student used a demonstration using tracing paper to justify the Exterior Angle Theorem, review previously established theorems about angle relationships, such as:
(1) a linear pair of angles is supplementary and (2) the measures of the interior angles of a triangle sum to 180°. Then ask the student to logically reason about the
relationship among the measures of
,
, and
.
Ask the student to draw a line through point C parallel to
and reason about the angle relationships formed by the intersection of a transversal (
or
).
Consider implementing other 8.G.1.5 tasks to further assess the student.
ACCOMMODATIONS & RECOMMENDATIONS
Special Materials Needed:
Justifying the Exterior Angle of a Triangle Theorem 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.8.G.1.5:
Description
Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles
created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example,
arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an
argument in terms of transversals why this is so.
page 5 of 5