Lesson Solved problems on angles of a polygon

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<H2>Solved problems on angles of a polygon</H2>

This lesson is a small collection of typical comparatively simple problems on finding angles of a polygon. 
The goal of this text is to teach you by examples to make first steps in this area.

<H3>Problem 1</H3>Four interior angles of a pentagon are of &nbsp;107°, &nbsp;93°, &nbsp;89° &nbsp;and &nbsp;92°, &nbsp;correspondingly. &nbsp;Find the measure of the fifth, &nbsp;missing angle.

<B>Solution</B>

The sum of interior angles of a pentagon is &nbsp;(5-2)*180° = 3*180° = 540°.

So, &nbsp;we need to distract the four known angle measures from &nbsp;540° &nbsp;to get the measure of the missing fifth angle:

540° - (107° + 93° + 89° + 92°)  = 159°.

<B>Answer</B>. &nbsp;The missed fifth angle of the pentagon has the measure of 159°.

<H3>Problem 2</H3>One angle of a pentagon is &nbsp;160°. &nbsp;Four of its other angles are congruent. &nbsp;Find the measures of the angles of the pentagon.

<B>Solution</B>

The sum of interior angles of a pentagon is &nbsp;(5-2)*180° = 3*180° = 540°.

So, &nbsp;we need to distract the known angle measure &nbsp;160° &nbsp;from &nbsp;540° &nbsp;and divide by &nbsp;4:

{{{(540-160)/4}}} = 95°.

<B>Answer</B>. &nbsp;The four other angles of the pentagon are of &nbsp;95° &nbsp;each.

<H3>Problem 3</H3>The angles of a &nbsp;6-sided polygon are in the ratio &nbsp;2:3:4:4:3:2. &nbsp;Find the angles measures.

<B>Solution</B>

The &nbsp;6&nbsp; angles of the hexagon are &nbsp;2x, &nbsp;3x, &nbsp;4x, &nbsp;4x, &nbsp;3x &nbsp;and &nbsp;2x, &nbsp;where &nbsp;x&nbsp; is the measure 
of the &nbsp;(now unknown)&nbsp; angle, &nbsp;which is the common measure of the given six angles, &nbsp;according to the condition.

Since the sum of interior angles of any hexagon is &nbsp;(6-2)*180° = 4*180° = 720°, you have an equation for the unknown measure x

2x + 3x + 4x + 4x + 3x + 2x = 720, &nbsp;&nbsp;&nbsp;&nbsp;or

18x = 720.

Hence, &nbsp;x = 40, &nbsp;and the common angle measure is &nbsp;40°.

Then the angles of the hexagon are of &nbsp;80°, &nbsp;120°, &nbsp;150°, &nbsp;150°, &nbsp;120° &nbsp;and &nbsp;80°.

<B>Answer</B>. &nbsp;The angles of the hexagon are of &nbsp;80°, &nbsp;120°, &nbsp;150°, &nbsp;150°, &nbsp;120° &nbsp;and &nbsp;80°.

<H3>Problem 4</H3>A polygon has two of its interior angles to be &nbsp;120° &nbsp;each and others are &nbsp;150° &nbsp;each. 
Calculate the number of sides of the polygon.

<B>Solution</B>

<pre>
2*120 + 150*(n-2) = 180*(n-2)     <<<---=== The sum of all interior angles of the polygon

240 = 180*(n-2) - 150*(n-2)

240 = 30*(n-2)   ====>  n-2 = {{{240/30}}} = 8   ====>  n = 8+2 = 10.   <U>Answer</U>.
</pre>

<H3>Problem 5</H3>A polygon has three of its interior angles to be &nbsp;120° &nbsp;each and others are &nbsp;150° &nbsp;each. 
Calculate the number of sides of the polygon.

<B>Solution</B>

<pre>
120*3 + (n-3)*150 = (n-2)*180.

Simplify and solve for n.

120*3 + 150n - 3*150 = 180n - 2*180

120*3 - 3*150 + 2*180 = 180n - 150n

270 = 30 n   ====>   n = {{{270/30}}} = 9.     <U>ANSWER</U>
</pre>

<H3>Problem 6</H3>Two angles of polygon are &nbsp;120° &nbsp;and &nbsp;90° &nbsp;and the rest are each &nbsp;166°. &nbsp;Find the number of the sides.

<B>Solution</B>

<pre>
120 + 90 + 166*(n-2) = 180*(n-2),

210 + 166n - 332 = 180n - 360,

238 = 14n,

N = 238/14 = 17 sides.
</pre>

<U>Answer</U>. 17 sides.
</pre>

My other introductory lessons on finding angles of triangles, parallelograms, quadrilaterals and polygons in this site are

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Points-lines-and-rays/Solved-problems-on-supplementary-and-complementary-angles-and-pairs-of-angles.lesson>Solved problems on supplementary and complementary angles</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Points-lines-and-rays/Solved-problems-on-angles-of-a-triangle.lesson>Solved problems on angles of a triangle</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Points-lines-and-rays/Solved-problems-on-angles-of-a-parallelogram.lesson>Solved problems on angles of a parallelogram</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Points-lines-and-rays/Solved-problems-on-angles-of-a-quadrilateral.lesson>Solved problems on angles of a quadrilateral</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Points-lines-and-rays/Solved-problems-on-missed-angle-of-a-polygon.lesson>Solved problems on missed angle of a polygon</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Points-lines-and-rays/Solved-problems-on-angles-of-a-regular-polygon.lesson>Solved problems on angles of a regular polygon</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Points-lines-and-rays/OVERVIEW-of-solved-problems-on-angles-of-triangles-quadrilaterals-and-polygons.lesson>OVERVIEW of solved problems on angles of triangles, parallelograms, quadrilaterals and polygons</A>

Use this file/link &nbsp;<A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-I - YOUR ONLINE TEXTBOOK</A>&nbsp; to navigate over all topics and lessons of the online textbook &nbsp;ALGEBRA-I.