Intent

Having learned that the shape of a triangle is determined by its side lengths, students now investigate whether a triangle is also determined by its angle measures.

Mathematics

If two triangles have the same angle measures, they have the same shape; that is, the two triangles are similar. (Because the sum of the angles in a triangle is fixed, only two angle measures need to be the same for one to draw this conclusion.) If, in addition to having the same angle measures, two triangles have sides connecting corresponding angles that are of the same length, then both the size and the shape of the triangles are the same.

Progression

Working in groups, students begin by drawing triangles with given angle measures and then compare their results. They repeat this for angle measures of their own choosing. Finally, they draw triangles with given angle measures and one given side length. They then talk about their work in a class discussion.

Approximate Time

40 minutes

Classroom Organization

Groups, followed by whole-class discussion

Materials

Protractors

Rulers

Doing the Activity

Students have seen, at least experimentally, that having corresponding sides proportional guarantees similarity for triangles. Now have them read and then do the activity in their groups.

Discussing and Debriefing the Activity

For Question 1, you might have some students trace their triangles onto transparencies. If so, choose examples of different sizes. Orient the transparencies so that the angles match up, superimposing the triangles. The nesting triangles will offer a vivid, visual confirmation of their similarity.

Students should recognize that there is a “rigidity” here, as they saw in Why Are Triangles Special? Although they can vary the size of the triangles in Question 1, the shape is determined.

In Question 3 students will see that once they choose one of the side lengths, they have no choice for the other two. (You might recognize this as the familiar ASA, or angle-side-angle, condition for triangle congruence.)

Post the general principle addressed in the activity:

If two triangles have their corresponding angles equal, then the triangles must be similar.

Some students may realize that only two pairs of corresponding angles need to be equal for similarity to exist, because the third angles are thus determined. If this comes up, ask students to explain their reasoning.