Primary Type: Formative Assessment
Status: Published
This is a resource from CPALMS (www.cpalms.org) where all educators go for bright ideas!
Resource ID#: 115763
Finding Angle Measures - 4
Students are asked to find the measure of an angle in a diagram containing two parallel lines and two transversals.
Subject(s): Mathematics
Grade Level(s): 9, 10, 11, 12
Intended Audience: Educators
Keywords: MFAS, parallel lines, transversal, angle bisector, supplementary angles, same-side interior angles
Resource Collection: CPALMS
ATTACHMENTS
MFAS_FindingAngleMeasures - 4_Worksheet.docx
MFAS_FindingAngleMeasures - 4_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 Finding Angle Measures - 4 worksheet.
2. The teacher asks follow-up questions, as needed.
Note: In the rubric, the following theorems are referenced by name:
Same­Side Interior Angles Theorem – If two parallel lines are intersected by a transversal, then same­side interior angles are supplementary.
Alternate Interior Angles Theorem - If two parallel lines are intersected by a transversal, then each pair of alternate interior angles is congruent.
TASK RUBRIC
Getting Started
Misconception/Error
The student is unable to correctly apply relevant theorems to find the missing angle measures.
Examples of Student Work at this Level
The student attempts to apply relevant theorems but does so incorrectly.
page 1 of 4 Questions Eliciting Thinking
Do you know what corresponding (or same-side interior or alternate interior) angles are?
Cover up transversal
and ask the student to identify a pair of corresponding angles and a pair of alternate interior angles.
Cover up transversal
and ask the student to identify a pair of same-side interior angles.
What does the Same-Side Interior Angles Theorem (or the Alternate Interior Angles Theorem) say?
What does it mean for an angle to be bisected? What is the relationship between the measures of
and
?
Instructional Implications
Review the definition of an angle bisector and guide the student to determine the relationship between the measures of
and
. Give the student opportunities
to find missing angle measures in diagrams involving bisected angles.
If needed, review the meaning of interior angles, corresponding angles, same-side interior angles, and alternate interior angles. Then guide the student through the
calculation of the measure of
providing justification. For example, review the Same-Side Interior Angles Theorem and emphasize the condition under which this
theorem can be applied (i.e., two parallel lines are intersected by a transversal). Assist the student in observing that there are two transversals in the diagram and ask the
student to draw
by extending
determine the measure of
. Then focus the student’s attention on transversal and assist the student in applying the Same-Side Interior Angles Theorem to
. Ask the student to apply the definition of an angle bisector to find the
.
Give the student opportunities to apply theorems related to special angle pairs formed by parallel lines and transversals in a variety of problem contexts.
Moving Forward
Misconception/Error
The student can find the missing angle measure but is unable to adequately justify his or her answers.
Examples of Student Work at this Level
The student correctly finds the measure of
. However, the student:
Provides the correct angle measure with minimal or no justification
Provides an explanation that contains errors or is significantly incomplete.
Describes the computations used without providing justification.
page 2 of 4 Questions Eliciting Thinking
What is the mathematical term used to describe
How do you know
and
and
? How do you know these angles are supplementary? What theorem supports this statement?
are congruent? What supports this statement?
What does it mean to justify your work? What postulates or theorems have you used in finding these angle measures?
Instructional Implications
Review the terms that apply to the angles and the angle relationships in the diagram and their definitions (e.g., angle bisector, supplementary angles, and same-side interior
angles). Review theorems that will be needed in the justifications (e.g., Same-Side Interior Angles Theorem). Explain that when justifying mathematical work, the student
should cite the relevant definitions, postulates, and theorems that support computational work. For example, model explaining since
and
, same-side interior angles
solve for
(e.g.,
and
+ 51 = 180
is a transversal of parallel lines
are supplementary by the Same-Side Interior Angles Theorem. Ask the student to use this fact to write an equation to
= 129°). Explain that since bisects
,
=
= 64.5° by the definition of an angle bisector.
Provide additional opportunities for the student to find missing angle measures using similar diagrams and to justify his or her work.
Almost There
Misconception/Error
The student is unable to correctly cite relevant definitions or theorems that support some aspect of his or her work.
Examples of Student Work at this Level
The student correctly calculates the measure of
and provides justification. However, the student is unable to correctly cite a relevant definition or theorem to
support some aspect of his or her work. For example, the student:
Applies the Alternate Interior Angles Theorem but does not explicitly state the conclusion. Additionally, the student does not justify subtracting 51 from 180.
Refers to alternate interior angles as “opposite” interior angles and is unable to cite the definition of an angle bisector to support dividing the measure of by two.
Questions Eliciting Thinking
Which pair of alternate interior angles is congruent? How do you know these angles are congruent?
What is the definition of an angle bisector? How does the definition support your work?
How do you know same-side interior angles are supplementary? Are they always supplementary?
Instructional Implications
Provide feedback to the student concerning any error or omission in his or her justification. Explain that a complete justification includes any relevant definitions, postulates,
or theorems that support conclusions drawn about angle measures or equations written to model angle relationships. Provide additional opportunities for the student to find
missing angle measures using similar diagrams and to justify his or her work.
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 applies the definition of an angle bisector and theorems related to parallel lines and transversals to correctly find the measure of
or her work by citing the relevant definitions and theorems. For example, the student writes:
and
are supplementary by the Same-Side Interior Angles Theorem. So,
is a transversal of parallel lines
+ 51 = 180 and
= 129°. Since and
. The student justifies his
, so same-side interior angles
bisects
,
=
=
64.5° by the definition of an angle bisector. Note: The student may not initially cite the relevant definition or theorem to justify each step but can do so upon request.
Questions Eliciting Thinking
How did you know to divide the measure of
by two?
page 3 of 4 What must be true of this diagram in order to use the Same-Side Interior Angles Theorem (or the Alternate Interior Angles Theorem)?
Why can’t all of the angles of triangle QRS equal 51°?
Instructional Implications
Challenge the student to find an alternate strategy for finding the measure of
.
Ask the student to prove the Same-Side Interior Angles Theorem.
Define alternate exterior angles and ask the student to prove that when two parallel lines are intersected by a transversal, alternate exterior angles are congruent.
ACCOMMODATIONS & RECOMMENDATIONS
Special Materials Needed:
Finding Angle Measures – 4 worksheet
SOURCE AND ACCESS INFORMATION
Contributed by: MFAS FCRSTEM
Name of Author/Source: MFAS FCRSTEM
District/Organization of Contributor(s): Okaloosa
Access Privileges: Public
License: CPALMS License - no distribution - non commercial
Related Standards
Name
MAFS.912.G-CO.3.9:
Description
Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include:
vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and
corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant
from the segment’s endpoints.
page 4 of 4