Unit 1, Lesson 13
Adding the Angles in a Triangle
Accelerated 7
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Sticky notes
The purpose of this activity is for students to recall properties and types of triangles and specific language for describing them. In this activity, students draw a variety of triangles and sort them into categories of their choosing. Monitor for students choosing categories that compare angle measures and side lengths. Students may use informal language to describe their categories, such as “all of the sides are different lengths” rather than “scalene.”
Select students who use these categories to share during the whole-class discussion:
Arrange students in groups of 4. Distribute 3 sticky notes per student. Give 1 minute of quiet work time for students to draw a triangle on each sticky note.
Tell each group to look at all of their triangles and sort them into categories. Give 2–3 minutes of group time followed by a whole-class discussion. Listen for language describing the side lengths and angles of triangles, such as “equilateral,” “right,” “obtuse,” and “acute.”
Invite previously selected students to share the categories they chose for the triangles. As students describe categories, organize their descriptions for all to see in a table or chart like this one:
| acute (all angles acute) | right (has a right angle) | obtuse (has an obtuse angle) | |
|---|---|---|---|
|
scalene (side lengths all different) |
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|
isosceles (at least two side lengths are equal) |
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|
equilateral (three side lengths equal) |
As students share, invite them to place their sticky note on the chart for all to see.
If any students drew triangles that fall under multiple categories, such as a right isosceles triangle, display it for all to see. If not, draw one for all to see. Ask students which categories the triangle belongs to. Students may describe an isosceles triangle as having two angles the same or two sides the same.
Ask students if they can draw any other triangles that fit into two categories. Examples include equilateral and acute, scalene and obtuse, scalene and right. Students may describe scalene as having all different side lengths or all different angles.
Ask students if it is possible to draw an equilateral triangle that has a right angle. (No, 3 right angles would not form a triangle, and all the angles are the same in an equilateral triangle.)
If time allows, continue filling out the table with an example of each type of triangle or an explanation of why that triangle is not possible.
MLR7: Compare and Connect
The purpose of this activity is for students to observe that three angles that total to 180 degrees can be used to create 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 create their own three angles from a line and check whether they can construct a triangle with their angles. As students observe the triangles of others in their group as well as the triangles created from their chosen angles, they use repeated reasoning to conclude that 3 angles decomposed from a straight line can form a triangle (MP8).
Monitor 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. Also monitor 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 necessary, demonstrate “making a triangle” before beginning the activity so students understand the intent. With three cut-out 60-degree angles, for example, an equilateral triangle can be built. 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.
Your teacher will give you a page with three sets of angles and a blank space. Cut out each set of three angles. Can you make a triangle from each set that has these same three angles?
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 them 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.
Here are some questions for discussion:
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 continue to investigate this idea 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 to each other sum to a line.
A
If we experiment with angles in a triangle, we find that the sum of the measures of the three angles in each triangle is
Through experimentation we find:
If we add the three angles of a triangle physically by cutting them off and lining up the vertices and sides, then the three angles form a straight angle.
If we have a line and two rays that form three angles added to make a straight angle, then there is a triangle with these three angles.
Help us improve by sharing suggestions or reporting issues.
Your teacher will give you a card with a picture of a triangle.
The measurement of one of the angles is labeled. Mentally estimate the measures of the other two angles.
Find two other students with triangles congruent to yours but with a different angle labeled. Confirm that the triangles are congruent, that each card has a different angle labeled, and that the angle measures make sense.
Enter the three angle measures for your triangle on the table your teacher has posted.