In this lesson, students examine sets of triangles in which all the triangles share 3 common measures of angles or sides. For example, suppose a triangle has angles that measure and and a side length that measures . Here are 3 triangles that have these measures:

This example shows 2 “different triangles” – the first two triangles are congruent, but the third is not, so it is different from the other two.

Students construct arguments and critique the reasoning of others as they decide whether triangles are congruent or not (MP3). 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.

Then students build on their observations of triangles with shared angle measures and side lengths, by drawing their own triangles with specified measures: a given angle, two given angles, and two given angles and a given side length. This helps students see the structure of certain triangles—namely that sometimes the given conditions allow only one possible triangle, sometimes more than one, and sometimes none (MP7). Students also gain experience using various tools to draw triangles with given conditions.

In this course, this lesson occurs after students learn what it means for two shapes to be congruent. Encourage students to use language about transformations to more precisely state why two shapes are congruent or not congruent instead of using the term “identical copies."