Sum and Measure of Angles in a Polygon

The angle measures of a nine-sided polygon form a sequence: $$\displaystyle (x+1)^\circ$$, $$(x+2)^\circ$$, $$(x+3)^\circ$$, etc. What is the measure of the largest angle?

Incorrect. You might have gotten this answer if you used $$\displaystyle 360^\circ \cdot (n-2)$$ for the sum of the angles. The actual formula is $$\displaystyle 180^\circ \cdot (n-2)$$. [[snippet]]

Incorrect. You might have gotten this answer if you used $$\displaystyle 180^\circ \cdot n$$ for the sum of the angles. The actual formula is $$\displaystyle 180^\circ \cdot (n-2)$$. [[snippet]]

Set the sum of the terms equal to the sum of the angles of the polygon and then solve for _x_. $$\displaystyle 9x+45 = 1{,}260$$ $$\displaystyle 9x = 1{,}215$$ $$\displaystyle x = 135$$ Since _x_ = 135, the largest angle is $$\displaystyle (x+9)^\circ = 144^\circ$$.

Incorrect. You might have gotten this answer if you found the _smallest_ angle in the polygon, instead of the _largest_ angle. [[snippet]]

Incorrect. You might have gotten this answer if you found the value of _x_. The problem actually asks for the largest angle in the polygon. [[snippet]]

That's right! The sum of the angles in an _n_-sided polygon is given in the formula $$\displaystyle 180^\circ \cdot (n-2)$$. Substitute 9 for _n_ into the formula to calculate the sum of the angles in the given polygon: $$\displaystyle 180^\circ \cdot (9-2) = 180^\circ \cdot 7 = 1{,}260^\circ$$. Then find the sum of the sequence in the problem by multiplying the average term by the number of terms. __Average__: In this sequence, the first and last terms are $$\displaystyle (x+1)^\circ$$ and $$(x+9)^\circ$$, so the average is $$\displaystyle \mbox{Average} = \frac{(x+1)+(x+9)}{2} = \frac{2x+10}{2} = x+5$$. __Number of terms__: There are nine terms in the sequence. __Sum of terms__: Multiply these two values. $$\displaystyle \mbox{Sum} = 9(x+5) = 9x+45$$

135º

136º

144º

175º

275º

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