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A Parallel Proof

Module by: Interactive Mathematics Program

Intent

Building on their work in More About Angles, students are ready to construct one of the traditional proofs of the angle sum property for triangles.

Mathematics

The sum of the angles in a triangle is 180°. (This fact relies on the parallel postulate for its truth.) The proof developed here, which uses students’ conjectures about the relationship between alternate interior angles formed by parallel lines cut by a transversal, is closely related to the investigation students did in Degree Discovery.

Progression

Students have been assuming that the sum of the angles in a triangle is 180°. They now work in groups to establish that this relationship is always true.

Approximate Time

25 minutes

Classroom Organization

Groups, followed by whole-class discussion

Doing the Activity

Have students work in groups on this activity. If a hint seems needed, you might ask, Which angles must be equal? Which angle sum is easy to find?

Discussing and Debriefing the Activity

After most groups seem to have found the proof, have one or two present their reasoning.

There are, basically, three steps to the argument.

  • x = s and y = t, because each is a pair of alternate interior angles formed by a transversal across parallel lines.
  • x + r + y = 180º, because these angles form a straight angle.
  • Substituting s for x and t for y gives s + r + t = 180º, as desired.

Key Questions

Which angles must be equal?

Which angle sum is easy to find?

Supplemental Activities

The Parallel Postulate (extension) offers students an opportunity to learn more about the history of the parallel postulate.

Exterior Angles and Polygon Angle Sums (extension) is an alternative proof of the angle sum property for triangles to the one in A Parallel Proof, based on the use of exterior angles.

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