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First recognize that the number of interior angles equals the number of exterior angles and
also equal the number of sides. So we can represent the number of sides of the convex polygon by
the letter N ... and N will also equal the number of interior angles and the number of
exterior angles.
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Just to make things a little easier, let's assume that the convex polygon is a regular polygon ...
meaning that all sides are equal, all interior angles are equal, and all exterior angles
are equal.
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One thing we know about polygons is that the sum of their exterior angles is equal to 360
degrees. Since we have assumed that the polygon is regular, we know that the measure of
each exterior angle is the same. Call this measure of each exterior angle mE. Since there
are N of these angles we can say that the sum of all exterior angles is N times mE and we
know that this sum is 360 degrees. Therefore, write the equation:
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N*mE = 360
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Solve this equation for N by dividing both sides by mE to get:
.
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Remember this equation for later use.
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And each interior angle is supplementary to an exterior angle. Therefore, the measure of
each interior angle (call this measure mI) equals 180 degrees minus the measure of the
exterior angle (mE). In equation form this is:
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mI = 180 - mE
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and since there are N of these interior angles we can multiply both sides by N to get that
the total or sum of the measures of the interior angles is:
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N*mI = N(180 - mE)
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But the problem tells you that the sum of the measures of the interior angles is 7740 degrees.
So we can say that:
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N(180 - mE) = 7740
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Solve this equation for N by dividing both sides by (180 - mE) to get:
.
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Now we have two equations for N ... the one we did earlier and this one. The two equations are:
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and
.
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Since the right sides of these two equations both equal N, the right sides must be equal to
each other. So we can write the equation:
.
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You can get rid of the denominators by multiplying both sides of this equation by
.
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When you do that multiplication, on the left side the mE in the denominator cancels the
mE in the multiplier. And the left side is then
. And on the right side
the 180 - mE in the denominator cancels the 180 - me in the multiplier and the left side
is then
. In equation form this is:
.
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If you multiply out the left side of this equation it becomes
.
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You can get rid of the -360*mE on the left side by adding 360*mE to both sides and you have:
.
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Now you can solve for mE by dividing both sides of the equation:
.
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by 8100 and you have:
.
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This tells you that the measure of each exterior angle is 8 degrees. You know that there
are N exterior angles, so the total of the measures of the exterior angles is 8*N degrees. But
the sum of the exterior angles must also equal 360 degrees. So you can set the two quantities
equal and you have:
.
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Solve for N by dividing both sides of this equation by 8 and the result is:
.
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This tells you that the polygon whose measures of the interior angles add up to 7740 degrees
is a polygon that has 45 sides.
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Assuming that the polygon was a regular polygon did not invalidate the solution. It just made
it a little easier to do and to see what was going on.
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Hope this helps you to understand the problem and how it could be solved.
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