Triangles with 3 Common Measures
In this lesson, students examine sets of triangles in which all the triangles share 3 common measures of angles or sides. Students learn to recognize when triangles are “identical copies” that are oriented differently on the page, and when they are different triangles (meaning triangles that are not identical copies). This prepares them for trying to draw more than one triangle given 3 measures in the next lesson.
For example, suppose a triangle has angles that measure \(A\) and \(B\) and a side length that measures \(x\). Here are 3 triangles that have these measures:
This example shows 2 “different triangles” (triangles that are not identical copies). The first two triangles are identical copies, so they are the same, but the third is not, so it is different than the other two.
Students see that the configuration of which sides and angles are adjacent to each other can help them decide whether triangles are identical copies or different triangles (not identical copies). In the example, the first two figures have angles \(A\) and \(B\) adjacent to side \(x\). However, in the third figure angle \(B\) is no longer adjacent to side \(x\). Here students can see that a good way to try to make a different triangle with the same 3 measures is to change which sides and angles are adjacent.
Students do not need to memorize how many different kinds of triangles are possible given different combinations of angles and sides, and they do not need to know criteria such as angle-side-angle for determining if two triangles are identical copies..
In IM 6–8 Math Accelerated, this lesson occurs after students learn what it means for two shapes to be congruent an that the sum of the interior angles of a triangle is 180 degrees. Encourage students to use the language they learned about transformations to more precisely state why two shapes are or are not congruent instead of using the term “identical copies”. Where applicable, also modify the suggested questions to use with students, such as those in the lesson synthesis, to use “congruent” instead of “identical copies.”
For example, when discussing the warm-up it is appropriate for students to use words like “reflection” or “rotation” when describing how they noticed the first set of triangles are all congruent. Similarly, students will formally study scaled copies in a later unit, so they are not likely to use the language of scaled copies when discussing the second set of triangles in the warm-up.
- Describe, compare, and contrast (orally and in writing) triangles that share three common measures of angles or sides.
- Justify (orally and using other representations) whether triangles are identical copies or are “different” triangles.
- Recognize that examining which side lengths and angle measures are adjacent can help determine whether triangles are identical copies.
Let’s contrast triangles.
- I understand that changing which sides and angles are next to each other can make different triangles.
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