Fact about triangles

A triangle is defined by three points, sometimes called vertices (plural of vertex) connected by three line segments, usually called sides.  Two sides meeting at a vertex create an angle, so a triangle also has three interior angles.

The interior angles of a triangle always have a sum of 180°.  This means that if two angles are given, the third can be found using subtraction.

Classification of Triangles.

One classification method for triangles is determined by the largest of the three angles.

If the largest angle is greater than 90°, the triangle is obtuse.


If the largest angle equals 90°, the triangle is right.


If the largest angle is less than 90°, the triangle is acute.

A second method of classification deals with how the angles relate to one another.

If at least two angles have the same measure, the triangle is isosceles.


If all angles have the same measure, the triangle is equilateral. (In this case, all angles are 60°.)

If all angles are different, the triangle is scalene.

An equilateral triangle is a special case of isosceles.

If a triangle is defined by side lengths instead of angles, it is easy to tell which classification is true.

All sides the same length means equilateral.

At least two lengths are the same means isosceles.

All different lengths means scalene.

The Triangle Inequality

If we choose three positive numbers at random, it might be that these cannot be the side lengths of a triangle.  Since these lengths are straight line segments and therefore the shortest distance between two points, the other two lengths have to add up to at least as much as the long length.  (Think of the three points.  If I travel from A to B, it cannot be a short cut if I instead travel from A to C to B.) Likewise, the difference between two distances has to be less than third distance.

If we let the three sides be labeled a, b  and c,  we can state the Triangle Inequality as follows.

a + b > c > |a - b|

If I pick three points at random on a plane, they might all be on the same line.  So if b is directly between a and c, we need to take that in account by changing the greater than signs into greater than or equal signs.
 
a + b >= c >= |a - b|