By definition, EVERY triangle has three sides that are connected. These three sides connect to form three angles. To learn more about triangles, we can study the relationships between the angles and sides of a given triangle. In fact, we can use the Triangle Inequality Theorem to study angles if we modify the theorem slightly.
Triangle Inequality Theorem for Angles: The largest angle in the triangle will be opposite the longest side, and the smallest angle in a triangle will be opposite the shortest side.
You can also reverse the two statements in the theory to arrive at its converse: The longest side in a triangle will be opposite the largest angle, and the shortest side in a triangle will be opposite the smallest angle.
Let's try some problems using what we know about angle-side relationships from the Triangle Inequality Theorem for Angles.
Problem 1
Problem 2
Question
What is the largest angle in triangle HJG? Which side is opposite that angle?
Question
What is the smallest angle in triangle HJG? Which side is opposite that angle?
Question
How can you order the sides from smallest to largest?
Largest = \(\small\mathsf{ \overline{GH}}\)
Next = \(\small\mathsf{ \overline{GJ}}\)
Smallest = \(\small\mathsf{ \overline{JH}}\)
Question
What is the longest side of triangle QPR? What is the angle opposite that side?
The angle opposite it is P.
Question
What is the shortest side of triangle QPR? What is the angle opposite that side?
Question
How might you summarize your answers?
The largest angle is P, and the smallest angle is R.
