standard_description: Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.
problem_activity: Activity
In later lessons, students will encounter situations in which they need to figure out if they have enough information to be sure two triangles are similar, based on angle measure alone. This activity introduces that concept as students grapple with whether the given information is enough to figure out that at least two pairs of corresponding angles are congruent.
Students will need to use their knowledge of the Triangle Angle Sum Theorem to figure out the third angle in each triangle.
Launch
Speaking, Reading: MLR5 Co-Craft Questions.
Use this routine to increase awareness of the language used to talk about the features of similar triangles. Before revealing the questions in this activity, display the image of both triangles. Ask students to write down possible mathematical questions that could be asked about the image. Invite students to compare their questions before revealing the actual questions. Listen for and amplify any questions about the features of both triangles. For example, “Are the triangles similar?”, “What are the values of
\(x\)
and
\(y\)
?”, and “What are the measures of the missing angles?”
Design Principle(s): Maximize meta-awareness; Support sense-making
Student Facing
Here are 2 triangles. One triangle has a 60 degree angle and a 40 degree angle. The other triangle has a 40 degree angle and an 80 degree angle.
[IMG]
Explain how you know the triangles are similar.
How long are the sides labeled
\(x\)
and
\(y\)
?
Student Response
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Student Facing
Are you ready for more?
Under what conditions is there an Angle-Angle Quadrilateral Similarity Theorem? What about an Angle-Angle-Angle Quadrilateral Similarity Theorem? Explain or show your reasoning.
Student Response
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Anticipated Misconceptions
If students are struggling unproductively, ask them to share what’s hard about this problem. (I’m stuck because I don’t have two pairs of corresponding angles.) Then ask them if it’s possible to find another corresponding angle using the information provided. (Yes, using the Triangle Angle Sum Theorem to figure out the third angle and check if the third pair of angles might be congruent.)
Activity Synthesis
The key idea is that the Triangle Angle Sum Theorem allows us to use the Angle-Angle Triangle Similarity Theorem even when we haven’t been given two pairs of congruent corresponding angles. Invite students to explain how they could tell the triangles were similar.


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