So, you were trying to be an excellent test taker and exercise for the GRE through PowerPrep virtual. Buuuut then you had actually some questions around the quant section—especially question 13 of the second Quantitative section of Practice Test 1. Those concerns experimentation our expertise of Polygons can be sort of tricky, yet never are afraid, has actually got your back!

Survey the Question

Let’s search the trouble for hints as to what it will be testing, as this will help shift our minds to think around what type of math understanding we’ll use to resolve this question. Pay attention to any type of words that sound math-specific and also anypoint one-of-a-kind around what the numbers look choose, and also mark them on our paper.

You are watching: The figure shows a regular 9 sided polygon

Let’s store what we’ve learned about this skill at the tip of our minds as we strategy this question.

What Do We Know?

Let’s closely check out via the question and make a list of the things that we understand.

We have a continual $9$-sided polygonWe desire to recognize the worth of an exterior angle to that polygon displayed in the figure

Develop a Plan

We understand that the sum of angles on one side of a straight line is $180°$ from the number, we have the right to see that if we deserve to discover the value of the inner angle at one vertex of the polygon, then we can subtract that value from $180°$ to acquire the worth of $x$.

To find the inner angle of any type of polygon, we have the right to divide it into triangles, understanding that all triangles have internal angles that sum as much as $180°$. Then multiply the variety of triangles by $180°$ and also finally divide by the variety of vertices of the polygon to acquire the value of its inner angle. This won’t be as challenging as it sounds, particularly once we begin illustration the triangles on our figure.

Solve the Question

First, let’s draw triangles founding at one vertex in our figure, favor this:

So below we deserve to watch that the amount of every one of the inner angles in our polygon deserve to be stood for as seven triangles. To discover the value of an internal angle within this polygon, we can simply multiply the variety of triangles by $180°$, then divide by the number of internal angles, which is nine.

$Interior Angle of a Polygon$$=$$180°·Number of Triangles/Number of Vertices$
$ $$ $
$Interior Angle of a Polygon$$=$$180°·7/9$
$ $$ $
$Interior Angle of a Polygon$$=$$9·20°·7/9$
$ $$ $
$Interior Angle of a Polygon$$=$$20°·7$
$ $$ $
$Interior Angle of a Polygon$$=$$140°$

Excellent! So the inner angle of a $9$-sided polygon is $140°$. We can see that $x$ and one inner angle lie on the exact same side of a right line, so their sum need to be $180°$. So $x=180°-140°$, or $x=40°$.

The correct answer is $40°$.

What Did We Learn

Now we recognize precisely just how to find the interior angle for any type of consistent polygon. We deserve to simply divide it into triangles, obtain the complete amount of the internal angles of the polygon by multiplying the variety of triangles by $180°$, then separating this amount by the variety of vertices of the polygon (which is additionally equal to the variety of sides of the polygon).

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