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Diagonals

One property of all convex polygons has to do with the number of
diagonals that it has:

Every convex polygon with n sides has n(n-3)/2 diagonals.

With this formula, if you are given either the number of diagonals or the number
of sides, you can figure out the unknown quantity. Diagonals become useful in
geometric proofs when you may need to draw in extra
lines or
segments, such as diagonals.

Figure %: Diagonals of polygons

The figure with 4 sides, above, has 2 diagonals, which accords to the formula,
since 4(4-3)/2 = 2. The figure with 8 sides has twenty diagonals, since 8(8-
3)/2 = 20.

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Interior Angles

The interior angles of polygons follow certain patterns based on the number
of sides, too. First of all, a polygon with n sides has n vertices, and
therefore has n interior angles. The sum of these interior angles is equal to
180(n-2) degrees. Knowing this,
given all the
interior angle measures but one, you can always figure out the measure of the
unknown angle.

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Exterior Angles

An exterior angle on a polygon is formed by extending one of the sides of
the polygon outside of the polygon, thus creating an angle
supplementary to the interior angle at that
vertex.
Because of the congruence of vertical
angles, it doesn't matter which side is
extended; the exterior angle will be the same.

The sum of the exterior angles of any polygon (remember only convex polygons are
being discussed here) is 360 degrees. This is a result of the interior angles
summing to 180(n-2) degrees and each exterior angle being, by definition,
supplementary to its interior angle. Take, for
example, a triangle with three vertices of 50 degrees, 70 degrees, and 60
degrees. The interior angles sum to 180 degrees, which equals 180(3-2).
Because the exterior angles are supplementary to the interior angles, they
measure, 130, 110, and 120 degrees, respectively. Summed, the exterior angles
equal 360 degreEs.

A special rule exists for regular polygons: because they are
equiangular, the exterior angles are also congruent, so the measure of any
given exterior angle is 360/n degrees. As a result, the interior angles of a
regular polygon are all equal to 180 degrees minus the measure of the exterior
angle(s).

Notice that the definition of an exterior angle of a polygon differs from
that of an exterior angle in a
plane. A polygon's exterior angle is not
equal to 360 degrees minus the measure of the interior angle. A polygon's
interior and exterior angles at a given vertex don't span the entire plane, they
only span half the plane. That is why they are supplementary--because their
measures sum to 180 degrees instead of 360.