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Please help with these geometry question that have to due with polygons:

  • 1.Which of the following are possible measures of the exterior angles of a polygon and how many sides does the polygon have: 90, 80, 75, 30, 46, 36, 2

    2.which of the following cant be the sum of the angle measures of a convex polygon: 1530, 3420, 6480 and 4500

    3 The measure of one angle of a octagon is is twice the other seven angles. What is the measure of each angle

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Please put each question in a different thread and outline your thoughts and work pertaining to each one, so the fine people of MSE don't tell you things you already know and can best help you understand the material. –  Clayton Feb 2 '13 at 14:39
    
Is this homework? If so, please use the homework tag. –  Parth Kohli Feb 2 '13 at 14:42

2 Answers 2

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The sum of all exterior angles in any polygon is $\mathbf{360^{\circ}}$. Thus, we have all the divisors of $360$ as a possible exterior angle as a choice and the number we have to divide $360$ by to get the divisor will be the number of sides in the particular polygon.

For example, we have $90$ as a divisor of $360$ so $90^{\circ}$ is a possible exterior angle. Since $90 \times 4 = 360$, the particular polygon with $90^{\circ}$ exterior angle will have $4$ sides.

Another fact: the sum of angles in a polygon with $n$ sides is $180(n - 2)$ degrees. So all the sums will be a multiple of $180$. Use that formula with $n = 8$ to get the sum of all interior angles in an octagon.

Remark: Technically, all the choices you have been provided with can be the measures of an exterior angle. If in the future, any question asks about regular polygons, you can predict that the answers will only be the divisors of $360$.

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$1.$ The sum of the measure of the exterior angles of a polygon is $360^\circ$

If it has uniform $n$ sides, each of the exterior angles will be $\frac{360^\circ}n$, so for uniform polygon, the value of each exterior angle must be a divisor of $360$ in degrees.

$2.$ The sum of angles of a convex polygon with $n$ sides is $180^\circ(n-2)$ which is clearly multiple of $180$ in degrees.

$3.$ The sum of The measure of angles of a octagon is $180^\circ(8-2)=1080^\circ$

Now, if the smaller angles is $\theta$ each, the larger will be $2\theta$

So, $2\theta+7\theta=1080^\circ$

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