The Student Will compare sides and angles among triangle

Prior
Knowledge: inequalities in one triangle; the triangle inequality; triangle congruence

Motivational
Question: Two sides of a given triangle are 8 and 10. Pick three possible values for the length of the third side, sketch each triangle, and compare and contrast the results.

Overview:

Use the motivational problem as an opportunity to review the triangle inequality and the one-triangle inequality theorems (largest side opposite largest angle etc).

The two-triangle theorems are:

SAS inequality
theorem

If two sides of one
triangle are congruent to two sides of another triangle, but the included angle
of the first triangle is larger than the included angle of the second triangle,
then the third side of the first triangle is longer than the third side of the
second triangle.

SSS Inequality
Theorem

If two sides of one
triangle are congruent to two sides of another triangle, but the third side of
the first triangle is longer than the third side of the second triangle, then the
included angle of the first triangle is longer than the included angle of the
second triangle.
An effective visualization of all the related theorems can be had in the following way: Assume that you are given two sides of the triangle, AB and BC, with known, fixed length. Keeping AB in place, rotate BC about the point B--that is, make a circle centered at B with radius BC. As point C moves, you can see the angles and sides getting longer (shorter) simultaneously, and you can also visualize the constraints of the triangle inequality in the two places BC is collinear with AB.

Key
Questions: How are angles and sides related in triangles? How do the SAS and SSS inequality theorems relate to their congruence counterparts?

Looking
Ahead: The above visualization will also be helpful in working with the ambiguous case in the Law of Sines.