standard_description: Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; 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.
problem_activity: Activity
The purpose of this activity is to decide if there is a sequence of translations, rotations, and reflections that take one figure to another and, if so, to produce one such sequence. Deciding whether or not such a sequence is possible uses the knowledge that translations, rotations, and reflections do not change side lengths or angle measures. The triangles
\(ABC\)
and
\(CFG\)
form part of a large pattern of images of triangle
\(ABC\)
that will be examined more closely in future lessons.
Monitor for students who use different transformations to take triangle
\(ABC\)
to triangle
\(CFG\)
and select them to share during the discussion. (There are two possible sequences in the Possible Responses section, but these are not the only two.)
Digital
Launch
Provide access to geometry toolkits. Give students 4 minutes quiet work time, 2 minutes to discuss with partner, and then time for a whole-class discussion.
If using the digital activity, have a brief discussion of the previous activity to highlight the transformations that the students used. Then give students 4 minutes of individual work time, 2 minutes to discuss with a partner, and then time for a whole-class discussion.
Action and Expression: Develop Expression and Communication
. Invite students to talk about their ideas with a partner before writing them down. Display sentence frames to support students when they explain their ideas. For example, “Triangle _____ is a rigid transformation of Triangle
\(ABC\)
because...,” “I agree/disagree because...,” or “Another transformation is _____ because….”
Supports accessibility for: Language; Organization
Conversing: MLR2 Collect and Display.
As students discuss their work with a partner, listen for and collect the language students use to describe each transformation. Record students’ words and phrases on a visual display (e.g., "Rotate triangle
\(ABC\)
90 degrees counterclockwise around point
\(C\)
," "Translate triangle
\(ABC\)
7 units right," etc.), and update it throughout the remainder of the lesson. Remind students to borrow language from the display as needed. This will help students read and use mathematical language during paired and whole-group discussions.
Design Principle(s): Optimize output (for explanation); Maximize meta-awareness
Student Facing
Here is a grid showing triangle
\(ABC\)
and two other triangles.
You can use a
rigid transformation
to take triangle
\(ABC\)
to
one
of the other triangles.
Which one? Explain how you know.
Describe a rigid transformation that takes
\(ABC\)
to the triangle you selected.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Print
Launch
Provide access to geometry toolkits. Give students 4 minutes quiet work time, 2 minutes to discuss with partner, and then time for a whole-class discussion.
If using the digital activity, have a brief discussion of the previous activity to highlight the transformations that the students used. Then give students 4 minutes of individual work time, 2 minutes to discuss with a partner, and then time for a whole-class discussion.
Action and Expression: Develop Expression and Communication
. Invite students to talk about their ideas with a partner before writing them down. Display sentence frames to support students when they explain their ideas. For example, “Triangle _____ is a rigid transformation of Triangle
\(ABC\)
because...,” “I agree/disagree because...,” or “Another transformation is _____ because….”
Supports accessibility for: Language; Organization
Conversing: MLR2 Collect and Display.
As students discuss their work with a partner, listen for and collect the language students use to describe each transformation. Record students’ words and phrases on a visual display (e.g., "Rotate triangle
\(ABC\)
90 degrees counterclockwise around point
\(C\)
," "Translate triangle
\(ABC\)
7 units right," etc.), and update it throughout the remainder of the lesson. Remind students to borrow language from the display as needed. This will help students read and use mathematical language during paired and whole-group discussions.
Design Principle(s): Optimize output (for explanation); Maximize meta-awareness
Student Facing
Here is a grid showing triangle
\(ABC\)
and two other triangles.
[IMG]
You can use a
rigid transformation
to take triangle
\(ABC\)
to
one
of the other triangles.
Which one? Explain how you know.
Describe a rigid transformation that takes
\(ABC\)
to the triangle you selected.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Student Facing
Are you ready for more?
A square is made up of an L-shaped region and three transformations of the region. If the perimeter of the square is 40 units, what is the perimeter of each L-shaped region?
[IMG]
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Activity Synthesis
Ask a student to explain why triangle
\(ABC\)
cannot
be taken to triangle
\(DBE\)
. (We are only using rigid transformations and therefore the corresponding lengths have to be equal and they are not.) If a student brings up that they think triangle
\(DBE\)
is a scale drawing of
\(ABC\)
, bring the discussion back to translations, rotations, and reflections, rather than talking about how or why triangle
\(DBE\)
isn’t actually a scale drawing of
\(ABC\)
.
Offer as many methods for transforming triangle
\(ABC\)
as possible as time permits, selecting previously identified students to share their methods. Include at least two different sequences of transformations. Make sure students attend carefully to specifying each transformation with the necessary level of precision. For example, for a rotation, that they specify the center of rotation, the direction, and the angle of rotation.
If time allows, consider asking the following questions:
"Can triangle
\(ABC\)
be taken to triangle
\(CFG\)
with only a translation?" (No, since
\(CFG\)
is rotated.)
"What about with only a reflection?" (No, because they have the same orientation.)
"What about with a single rotation?" (The answer is yes, but this question does not need to be answered now as students will have an opportunity to investigate this further in a future lesson.)


aligns: no