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Arrange students in groups of 2–4. Display the images for all to see. Give students 1 minute of quiet think time and ask them to indicate when they have noticed three images that go together and can explain why. Next, tell students to share their response with their group and then together to find as many sets of three as they can.
Which three go together? Why do they go together?
Invite each group to share one reason why a particular set of three go together. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Since there is no single correct answer to the question of which three go together, attend to students’ explanations and ensure that the reasons given are correct.
During the discussion, prompt students to explain the meaning of any terminology they use, such as “right angle,” “acute angle,” and “obtuse angle” and to clarify their reasoning as needed. Consider asking:
In this activity, students continue creating a triangle from given conditions and seeing if it will match a given triangle. This activity transitions from students just noticing things about triangles already drawn to students drawing triangles themselves to test whether conditions result in unique triangles.
As students work on the task, monitor for students who draw triangles that differ from those of their classmates.
Arrange students in groups of 2. Give students 3–5 minutes of quiet work time, followed by time to check with their partner. Discuss whether any of the triangles they drew are identical copies. Follow with a whole-class discussion. Provide access to geometry toolkits that include a ruler marked with centimeters and a protractor.
Three students have each drawn a triangle. For each description:
Jada’s triangle has one angle measuring
Andre’s triangle has one angle measuring
Lin’s triangle has one angle measuring
Students may have trouble recognizing that Lin’s triangle could have the pieces described in different orders. They are likely to immediately think of the side being between the two angles and not visualize other arrangements. Remind students of the task from the previous day and some of the triangles they saw there.
Select previously identified students to share their triangles.
To highlight the fact that there could be different triangles drawn, ask:
If not mentioned by students, explain that it could be possible that we all drew identical copies for Lin’s triangle (because it is most straightforward to draw the 5-cm side in between the 75° and 45° angles). However, that does not mean that we were given enough information about Lin’s triangle to draw an identical copy of it. The problem did not say that we had to put the 5-cm side between those two angles.
Display the image of Lin’s triangle for all to see. Invite students to confirm that it matches the description of Lin’s triangle. Ask whether any student drew an identical copy of Lin’s triangle.
Introduce the word “unique.” Explain to students that in all three cases, the information given is not enough to determine a unique triangle, not even for Lin’s triangle, because there is more than 1 way we can draw a triangle with those given conditions. Ask students, “What information would Lin have to give us to make the triangle unique (so we knew our drawing would be an identical copy of her triangle)?”
Before moving on to the next activity, it would be helpful to model how Lin drew her triangle:
Sometimes, we are given two different angle measures and a side length, and it is impossible to draw a triangle. For example, there is no triangle with side length 2 and angle measures
In the figure a horizontal line segment is drawn and labeled 2. On the left end of the line segment, a dashed line is drawn upward and to the left. The angle formed between the dashed line and the horizontal line is labeled 120 degrees. On the right end of the horizontal line, a dashed line is drawn upward and to the right. The angle formed between the dashed line and horizontal line is labeled 100 degrees.
Sometimes, we are given two different angle measures and a side length between them, and we can draw a unique triangle. For example, if we draw a triangle with a side length of 4 between angles
Any triangle drawn with these three conditions will be identical to the one above, with the same side lengths and the same angle measures.
Keep students in the same groups. Tell students they must try at least two different times to draw a triangle with the measurements given in each problem. Give students 5 minutes of quiet work time followed by time to discuss their different triangles with a partner. Follow with a whole-class discussion. Provide access to geometry toolkits.
Draw as many different triangles as you can with each of these sets of measurements:
Some students may draw two different orientations of the same triangle for the third set of conditions, with the 4-cm side in between the
Some students may say the third set of measurements determines one unique triangle, because they assume the side length must go between the two given angle measures. Remind them of the discussion about Lin’s triangle in the previous activity.