Lesson 13

Students try to draw triangles satisfying different properties. They complete the table and then check with a partner whether or not they agree that the pictures are correct or that no such triangle can be drawn. The goals of this warm-up are:

• Reviewing different properties and types of triangles.
• Focusing on individual angle measures in triangles in preparation for studying their sum.

Note that we use the inclusive definition of isosceles triangle having at least two congruent sides. It is possible that in students’ earlier experiences, they learned that an isosceles triangle has exactly two congruent sides. This issue may not even come up, but be aware that students may be working under a different definition of isosceles than what is written in the task statement.

Arrange students in groups of 2. Quiet work time for 3 minutes to complete the table followed by partner and whole-class discussion.

Invite students to share a few triangles they were able to draw such as:

• Right and isosceles
• Equilateral and acute

Ask students to share which triangles they were unable to draw and why. For example, there is no right equilateral or obtuse equilateral triangle because the side opposite the right (or obtuse) angle is longer than either of the other two sides. 

This is a matching activity where each student receives a card showing a triangle and works to form a group of three. Each card has a triangle with the measure of only one of its angles given. Students use what they know about transformations and estimates of angle measures to find partners with triangles congruent to theirs. Each unique triangle’s three interior angles are then displayed for all to see. Students notice that the sum of the measures of the angles in each triangle is 180 degrees.

During this activity, students can use MP7, thinking about applying rigid motions to their triangle to see if it might match up with another student's triangle. Or they may identify that their triangle is acute, right, or obtuse and use this as a criterion when they search for a congruent copy (also MP7).

You will need the Find All Three blackline master for this activity.

Provide access to geometry toolkits. Distribute one card to each student, making sure that all three cards have been distributed for each triangle. (If the number of students in your class is not a multiple of three, it’s okay for one or two students to take ownership over two cards showing congruent triangles.) Explain that there are two other students who have a triangle congruent to theirs that has been re-oriented in the coordinate plane through combinations of translations, rotations, and reflections. Instruct students to look at the triangle on their card and estimate the measures of the other two angles. With these estimates and their triangle in mind, students look for the two triangles congruent to theirs with one of the missing angles labeled.

Prepare and display a table for all to see with columns angle 1, angle 2, angle 3 and one row for each group of three students. It should look something like this:

Once the three partners are together, they complete one row in the posted table for their triangle’s angle measures. Whole-class discussion to follow.

Students might ask if they can use tracing paper to find congruent triangles. Ask how they would use it and listen for understanding of transformations to check for congruence. Respond that this seems to be a good idea.

Begin the discussion by asking students:

• "What were your thoughts as you set about to find your partners?"
• "How did you know that you found a correct partner?"

Expect students to discuss estimates for angle measures and their experience of how different transformations influence the position and appearance of a polygon.

Next look at the table of triangle angles and ask students:

• "Is there anything you notice about the combinations of the three angle measures?"
• "Is there something in common for each row?"

Guide students to notice that the sum in each row is the same, 180 degrees. Ask whether they think this is always be true for any triangle. Share with students that in the next activity, they will work towards considering whether this result is true for all triangles.

In the optional activity, students found that the sum of the angles of all the triangles on the cards was 180 degrees and questioned if all triangles have the same angle sum. In this activity, students experiment with the converse: If we know the measures of three angles sum to 180 degrees, can these three angles be the interior angles in a triangle?

Students cut out three angles that form a line, and then try to use these three angles to make a triangle. Students also get to create their own three angles from a line and check whether they can construct a triangle with their angles. 

Watch for students who successfully make triangles out of each set of angles and select them to share (both the finished product and how they worked to arrange the angles) during the discussion. Watch also for how students divide the blank line into angles. It is helpful if the rays all have about the same length as in the pre-made examples.

Arrange students in groups of 4. Provide access to geometry toolkits, especially scissors. Distribute 1 copy of the black line master to each group.

Instruct students to cut the four individual pictures out of the black line master. Each student will work with one of these. Instruct the student with the blank copy to use a straightedge to divide the line into three angles (different from the three angles that the other students in the group have). Demonstrate how to do this if needed. 

If needed, you may wish to demonstrate “making a triangle”  part of the activity so students understand the intent. With three cut-out 60 degree angles, for example, you can build an equilateral triangle. Here is a picture showing three 60 degree angles arranged so that they can be joined to form the three angles of an equilateral triangle. The students will need to arrange the angles carefully, and they may need to use a straightedge in order to add the dotted lines to join the angles and create a triangle.

If time allows, have students do a "gallery walk" at the start of the discussion. Ask students to compare the triangle they made to the other triangles made from the same angles and be prepared to share what they noticed. (For example, students might notice that all the other triangles made with their angles looked pretty much the same, but were different sizes.) If students do not bring it up, direct students to notice that all of the "create three of your own angles" students were able to make a triangle, not just students with the ready-made angles.

Ask previously selected students to share their triangles and explain how they made the triangles. To make the triangles, some trial and error is needed. A basic method is to line up the line segments from two angles (to get one side of the triangle) and then try to place the third angle so that it lines up with the rays coming from the two angles already in place. Depending on the length of the rays, they may overlap, or the angles may need to be moved further apart. Ask questions to make sure that students see the important connection:

• "How do you know the three angles you were given sum to 180 degrees?" (They were adjacent to each other along a line.)
• "How do you know these can be the three angles of a triangle?" (We were able to make a triangle using these three angles.)
• "What do you know about the three angles of the triangle you made and why?" (Their measures sum to 180 because they were the same three angles that made a line.)

Ask students if they think they can make a triangle with any set of three angles that form a line and poll the class for a positive or negative response. Tell them that they will investigate this in the next lesson and emphasize that while experiments may lead us to believe this statement is true, the methods used are not very accurate and were only applied to a few sets of angles. 

If time permits, perform a demonstration of the converse: Start with a triangle, tear off its three corners, and show that these three angles when placed adjacent each other sum to a line.