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Triangle Inequalities

Triangles are governed by two important inequalities. The first is often
referred to as the triangle inequality. It states that the length of a side of
a triangle is always less than the sum of the lengths of the other two sides.
Can you see why this must be true? Were one side of a triangle longer than the
sum of the lengths of the other two, the triangle could not exist. As one side
grows, the other two collapse toward that side until the
altitude from the vertex opposite the growing
side eventually becomes zero. This (an altitude of zero) would happen if the
length of the one side was *equal* to the sum of the lengths of the other
two. For this reason, the length of any side must be less than the sum of the
lengths of the other sides.

The second inequality involving triangles has to do with opposite angles and
sides. It states that when a pair of angles are unequal, the sides opposite
them are also unequal. The converse is true also: when a pair of sides are
unequal, so are their opposite angles. In essence, this theorem complements the
theorem involving isosceles triangles,
which stated that when sides or angles were equal, so were the sides or angles
opposite them. The theorem about unequal pairs, though, goes a little farther.
Given unequal angles, the theorem holds that the longer side of the triangle
will stand opposite the larger angle, and that the larger angle will stand
opposite the longer side. This inequality is helpful to prove triangles
*aren't* congruent.

Figure %: The larger of two unequal angles is opposite the longer of two unequal
sides,
and vice versa.

Notice the symbols in the figure above. When angles or sides are equal, the
same number of tick marks, or small dashes, can be drawn on them. In a case
where sides or angles are unequal, this can be symbolized by different numbers
of tick marks on the angles or sides. More tick marks signifies a greater
measure.

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Exterior Angles of a Triangle

A triangle's exterior angle is just like that of any polygon; it is the angle
created when one side of the triangle is extended past a vertex. The exterior
angle has two interesting properties that follow from one another. 1) The
exterior angle at a given vertex is equal in measure to the sum of the two
remote interior angles. These remote interior angles are those at the other
two vertices of the triangle. 2) Knowing this, it follows that the measure of
any exterior angle is always greater than the measure of either remote interior
angle. The first fact (1), the equality, is useful for proving congruence; the
second fact (2), the inequality, is useful for disproving congruence.

Figure %: Angle 4 is greater than angle 2 and angle 3; angle 4 = angle 2 + angle
3