The Student Will investigate and apply geometric inequalities in triangles

The Student Will prove geometric inequalities using Indirect Proof.

Prior
Knowledge: Basic Triangle Geometry; Properties of Inequalities; Proof by Contradiction; Exterior Angle Theorem

Motivational
Question: State and prove the Exterior Angle Theorem. Make a conjecture about what the Exterior Angle Inequality Theorem might say.

Overview:

The following theorems should be investigated and proved:

Exterior Angle Inequality Theorem

Longer Side Opposite Larger Angle Theorem

Larger Angle Opposite Longer Side Theorem

The perpendicular line is the shortest distance between a point and a line (or plane)

The proof of the Exterior Angle Inequality Theorem is a consequence of the Exterior Angle Theorem and a property of inequality.

One way to prove the Longer Side Opposite Larger Angle Theorem is by drawing an appropriate cevian that makes an internal isosceles triangle and then applying the Exterior Angle Inequality.
To prove the Larger Angle Opposite Longer Side Theorem, use indirect proof and the trichotomy property of inequality.
The last theorem is of great importance and is a direct corollary of the Larger Angle Opposite the Longer Side theorem (a triangle can have, at most, one right or obtuse angle).

Key
Questions: Why is the Exterior Angle Inequality Theorem true? How can we apply the Opp. Angle/ Opp Side Theorems? Why is the perpendicular the shortest distance between a point and a line/plane? Can we establish that fact in another way?

Looking Ahead: These theorems will establish the Triangle Inequality. Also, when students reach trigonometry, these theorems will help to provide reality checks when solving triangles--i.e., does it make sense that the angle opposite the longest side is 54 degrees?, etc.