standard_description: Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.
problem_activity: Activity
Students continue to practice drawing triangles from given conditions and categorizing their results. This activity focuses on the inclusion of a single angle and two sides. Again, they do not need to memorize which conditions result in unique triangles, but should begin to notice how some conditions (such as the equal side lengths) result in certain requirements for the completed triangle.
There is an optional blackline master that can help students organize their work at trying different configurations of the first set of measurements. If you provide students with a copy of the blackline master, ask them to determine whether any of the configurations result in the same triangle, as well as whether any one configuration results in two possible triangles.
Digital
Launch
Keep students in same groups. Remind students of the activity in a previous lesson where they used the strips and fasteners to draw triangles on their paper. Ask what other tool also helps you find all the points that are a certain distance from a center point (a compass). Distribute optional blackline masters if desired. Provide access to geometry toolkits and compasses.
Give students 7–8 minutes of partner work time, followed by a whole-class discussion.
If students have access to digital activities there is an applet that allows for triangle construction.
Reading, Speaking: MLR5 Co-craft Questions.
Display just the statement: “One angle measures 40 degrees, one side measures 4 cm, and one side measures 5 cm.” Invite students to write down possible mathematical questions that could be asked with this information. Ask students to compare the questions generated with a partner before selecting 1–2 groups to share their questions with the class. Listen for the ways the given conditions are used or referenced in students’ questions. This routine will help students to understand the context of this problem before they are asked to create a drawing.
Design Principle(s): Maximize meta-awareness; Cultivate conversation
Student Facing
Use the applet to draw triangles.
Draw as many different triangles as you can with each of these sets of measurements:
One angle measures
\(40^\circ\)
, one side measures 4 cm and one side measures 5 cm.
Two sides measure 6 cm and one angle measures
\(100^\circ\)
.
Did either of these sets of measurements determine one unique triangle? How do you know?
Student Response
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Print
Launch
Keep students in same groups. Remind students of the activity in a previous lesson where they used the strips and fasteners to draw triangles on their paper. Ask what other tool also helps you find all the points that are a certain distance from a center point (a compass). Distribute optional blackline masters if desired. Provide access to geometry toolkits and compasses.
Give students 7–8 minutes of partner work time, followed by a whole-class discussion.
If students have access to digital activities there is an applet that allows for triangle construction.
Reading, Speaking: MLR5 Co-craft Questions.
Display just the statement: “One angle measures 40 degrees, one side measures 4 cm, and one side measures 5 cm.” Invite students to write down possible mathematical questions that could be asked with this information. Ask students to compare the questions generated with a partner before selecting 1–2 groups to share their questions with the class. Listen for the ways the given conditions are used or referenced in students’ questions. This routine will help students to understand the context of this problem before they are asked to create a drawing.
Design Principle(s): Maximize meta-awareness; Cultivate conversation
Student Facing
Draw as many different triangles as you can with each of these sets of measurements:
One angle measures
\(40^\circ\)
, one side measures 4 cm, and one side measures 5 cm.
Two sides measure 6 cm, and one angle measures
\(100^\circ\)
.
Did either of these sets of measurements determine one unique triangle? How do you know?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Anticipated Misconceptions
Some students may draw two different orientations of the same triangle for the first set of conditions, with the
\(40^\circ\)
angle in between the 4 cm and 5 cm sides. Prompt them to use tracing paper to check whether their two triangles are really different (not identical copies).
If students struggle to create more than one triangle from the first set of conditions, prompt them to write down the order they already used for their measurements and then brainstorm other possible orders they could use.
Activity Synthesis
Ask one or more students to share how many different triangles they were able to draw with each set of conditions. Select students to share their solutions.
If not brought up in student explanations, point out that for the first problem, one possible order for the measurements (
\(40^\circ\)
, 5 cm, 4 cm) can result in two different triangles (the bottom two in the solution image). One way to show this is to draw a 5 cm segment and then use a compass to draw a circle with a 4 cm radius centered on the segment’s left endpoint. Next, draw a ray at a
\(40^\circ\)
angle centered on the segment’s right endpoint. Notice that this ray intersects the circle twice. Each one of these points could be the third vertex of the triangle. While it is helpful for students to notice this interesting aspect of their drawing, it is
not
important for students to learn rules about the number of possible triangles given different sets of conditions.
If the optional blackline master was used, ask students:
“Which configurations made identical triangles?” (the top left and bottom left)
“Which configurations made more than one triangle?” (the bottom right)
If not mentioned by students, explain to students that the top left and bottom left configurations result in the same triangle, because in both cases the
\(40^\circ\)
angle is in between the 4 cm and 5 cm sides and that the bottom right configuration results in two different triangles, because the arc intersects the ray in two different places.
MLR 1 (Stronger and Clearer Each Time): Before discussing the second set of conditions as a whole class, have student pairs share their reasoning for why there were no more triangles that could be drawn with the given measures, with two different partners in a row. Have students practice using mathematical language to be as clear as possible when sharing with the class, when and if they are called upon.
Engagement: Develop Effort and Persistence.
Break the class into small group discussion groups and then invite a representative from each group to report back to the whole class.
Supports accessibility for: Attention; Social-emotional skills


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