What could possibly be true?[br]
What definitely can’t be true?[br]
Player 1: You are the transformer. Take the transformer card.[br][br]Player 2: Select a triangle card. Do not show it to anyone. Study the diagram to figure out which sides and which angles correspond. Tell Player 1 what you have figured out.[br][br]Player 1: Take notes about what they tell you so that you know which parts of their triangles correspond. Think of a sequence of rigid motions you could tell your partner to get them to take one of their triangles onto the other. Be specific in your language. The notes on your card can help with this.[br][br]Player 2: Listen to the instructions from the transformer. Use tracing paper to follow their instructions. Draw the image after each step. Let them know when they have lined up 1, 2, or all 3 vertices on your triangles.
Replay invisible triangles, but with a twist. This time, the transformer can only use reflections—the last 2 sentence frames on the transformer card. You may wish to include an additional sentence frame: Reflect _____ across the angle bisector of angle _____.
[table][tr][td][list][*][math]\overline{AB}\cong\overline{DE}[/math][/*][*][math]\overline{AC}\cong\overline{DF}[/math][/*][*][math]\overline{BC}\cong\overline{EF}[/math][/*][*][math]\angle A\cong\angle D[/math][/*][*][math]\angle B\cong\angle E[/math][/*][*][math]\angle C\cong\angle F[/math][/*][/list][/td][td][img]data:image/png;base64,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[/img][/td][/tr][/table][br]Here are the steps Noah had to tell Priya to do before all 3 vertices coincided:[br][list][*]Translate triangle [math]ABC[/math] by the directed line segment from [math]A[/math] to [math]D[/math].[/*][*]Rotate the image, triangle [math]A'B'C'[/math], using D as the center, so that rays [math]A'B'[/math] and [math]DE[/math] line up.[/*][*]Reflect the image, triangle [math]A''B''C''[/math], across line [math]DE[/math].[/*][/list][br]After those steps, the triangles were lined up perfectly. Now Noah and Priya are working on explaining why their steps worked, and they need some help. Answer their questions.[br][br]First, we translate triangle [math]ABC[/math] by the directed line segment from [math]A[/math] to [math]D[/math]. Point [math]A'[/math] will coincide with [math]D[/math] because we defined our transformation that way. Then, rotate the image, triangle [math]A'B'C'[/math], by the angle [math]B'DE[/math], so that rays [math]A''B''[/math] and [math]DE[/math] line up.[br][br][br]We know that rays [math]A''B''[/math] and [math]DE[/math] line up because we said they had to, but why do points [math]B''[/math] and [math]E[/math] have to be in the exact same place?
How do we know that now, the image of ray[math]A''C''[/math] and ray [math]DF[/math] will line up?[br]
How do we know that the image of point [math]C''[/math] and point [math]F[/math] will line up exactly?[br]