Triangle Inequality Theorems in One, and Two Triangles.

What is needed to do in order to find out if three line segments will form a triangle, or not? How may you order the sides of a triangle, if you know the angles? How may you order the angles, if you know the sides? If you have two triangles with two corresponding pairs of congruent sides, how do you know which one has greatest angle included to those two sides? or the largest side opposite to the included angle?

This lesson starts with simple steps showing you the triangle inequality theorems in one triangle, and then it moves to triangle inequality in two triangles. In the process you are given examples, and suggested problems to test your understanding. Go for it!

Lesson's Content

Lesson's Glossary

Included angle
The angle made by two consecutive sides of a polygon.

Included side
The side between two consecutive angles in a polygon.

Polygon
A polygon is a two-dimensional geometric figure with these characteristics: 
It is made of straight line segments.
Each segment touches exactly two other segments, one at each of its endpoints. 
It is closed -- it divides the plane into two distinct regions, one inside and the other outside the polygon.

Side of a polygon
A single segment from the union that forms a polygon.

Triangle
A polygon with three sides.

Triangle inequality
The triangle inequality says that for three lengths to make a triangle, the sum of the lengths of any two sides must be greater than the third length.

Interactive Geometric Applets: Relevant Theorems.

In one triangle, if you have two noncongruent sides, then opposite to the longer side

you have the angle with the greater measure; and opposite to the shorter side you

have the angle with the lesser measure. Use the applet shown below to test the theorem

and its converse.

Given a line and a point outside the line, the shortest distance between the two is the

length of a perpendicular sement connecting them. Interact with this applet to verify it.

Given a plane and a point not in the plane, the shortest distance between the two is given

by the length of a perpendicular segment connecting both. Drag the point around the

plane to verify it.

The Triangle Inequality Theorem states than taking two sides of a given triangle; their

sum is always greater than the value for the third remaining side.

Use the interactive applet to check this theorem.

Finally, you have that given two triangles, and two pairs of congruent sides in them; you have

SSS Inequality Theorem, and SAS Inequality Theorem (Hinge Theorem). The first states that opposite

to the third longer side, you have the included angle with the greater measure; and the second states

that opposite to the included angle with the greater measure you have the longer third side.

Use the below applet to visualize these inequality relationships.

Vocabulary Puzzle Interactive

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