Question 269205
i have had trouble working on this math problem and i was wondering if someone could help me? Please and Thank you! I would deeply appreciate it!

What is the ratio of the measure of an interior angle to the measure of an exterior angle in a regular hexagon? A regular decagon? A regular n-gon?
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Let's do the last one first:

The sum of the interior angles of any n-gon is given by

{{{Sum = "180°"(n-2)}}}

If the polygon is regular then all of its interior angles are equal in
measure, so each one of them must be that sum divided by n. So

{{{Each_interior_angle=("180°"(n-2))/n}}}

The sum of the exterior angles of any n-gon is 360°

If the polygon is regular then all of its exterior angles are equal in
measure, so each one of them must be 360° divided by n. So

{{{Each_exterior_angle=("360°")/n}}}

So the ratio of the measure of an interior angle to the measure of an 
exterior angle in a regular n-gon is

{{{(("180°"(n-2))/n)/("360°"/n)}}}

{{{(("180°"(n-2))/n)}}}{{{"÷")}}}{{{("360°"/n)}}}


Invert the second fraction and change division to multiplication

{{{(("180°"(n-2))/n)}}}{{{"×")}}}{{{(n/"360°")}}}

Cnacel the n's and the 180° into the 360° getting 2 on the bottom:

{{{(n-2)/2}}}

A regular hexagon has 6 sides, so substitute n=6 in the equation:

{{{(6-2)/2=4/2=2}}} and the ratio is 2 to 1 or 2:1.

A regular decagon has 10 sides, so substitute n=10 in the equation:

{{{(10-2)/2=8/2=4}}} and the ratio is 4 to 1 or 4:1.

Edwin</pre>