hellobuddy
if each interior angle of a regular polygon measures 144 degrees, how many sides does the polygon have?
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hellobuddy
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Please show steps on how to solve.
jesusisrisen
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obtuse angles but idont know about a polygon with obtuse angles
JamesJ
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Ok. What's the sum of angles of a triangle? 180 degrees. A triangle is a polygon with 3 sides.
What's the sum of the angles of a square? A square is a polygon with 4 sides. What's the answer here?
hellobuddy
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the answer is supposed to be 10 sides
how do you solve that?
jesusisrisen
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decagon?
JamesJ
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We'll get there ... follow me in logic first
hellobuddy
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ok
JamesJ
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What's the sum of angles of a square?
hellobuddy
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180 (4-2) =
360
Safiah
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360 ofcourse.as all angles are 90
Safiah
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the polygon in question has 10 sides
hellobuddy
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^ why?
JamesJ
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Right ... so you've got the formula for an n-sided polygon. It has a sum of angles
\[ 180(n-2) \]
and that's because you can make a polygon out of triangles. Stick two triangles together and you have a square, hence the sum of their angles is 180 + 180. Take a square add a triangle and you have a pentagon.|dw:1327439866826:dw|
hellobuddy
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i know how to do that, how do you solve when you dont know the number of sides?
JamesJ
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Now, if an n-sided polygon has a sum of angles 180(n-2), what's the size of the angles if each of them is the same? Well, there are n angles, hence one angle has the size of
\[ \frac{180(n-2)}{n} \]
JamesJ
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For your problem, set that equal to 144 and solve for n.
Safiah
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the general rule is
Each Angle (of a Regular Polygon) = (n-2) × 180° / n
so 144=(n-2)x180/n
144n=(n-2)x180
144n=180n-360
144n-180n=-360
36n=360
n=10
hellobuddy
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so, James J, i do 180 (n-2) / n = 144
144-180+2= n/n?
hellobuddy
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NOPE THATS NOT RIGHT ^
hellobuddy
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i dont understnad..
JamesJ
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Safiah has written this out for you. But to start it off again
\[ \frac{180(n-2)}{n} = 144 \]
hence
\[ 180(n-2) = 144n \]
Can you do it now?
hellobuddy
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OH WOW
I GET IT!
hellobuddy
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sooo 180( n-2) = 144n
wiat no, what do i do now?
JamesJ
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...hence
\[ 180n - 360 = 144n \]
Now?
Safiah
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expand the brackets hellobudy
hellobuddy
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nope.. now what?
JamesJ
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Subtract 144n from both sides...
\[ 180n - 144n - 360 = 0 \]
i.e.
\[ 36n - 360 = 0 \]
Now can you finish it?
hellobuddy
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-360 = -36
and then you divide 360 by 36
and you get ten
JamesJ
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right,
\[ 36n = 360 \]
hence
\[ n = \frac{360}{36} = 10 \]
hellobuddy
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k i think i get it..
hellobuddy
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thank you!
JamesJ
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Do yourself a favor. Take a blank piece of paper. Write out the solution again. When you can do that without looking at anything --such as this web or another version of the solution--then you know you understand the solution.
hellobuddy
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ok! can you give me another example so i can practice? use a different degrees?
hellobuddy
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ok ill try with 100 degrees
JamesJ
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No, that won't work. One sec.
JamesJ
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try 108 degrees
hellobuddy
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ALRIGHT :)
hellobuddy
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5 sides?
JamesJ
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yes
hellobuddy
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:D
JamesJ
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last one, 162 degrees
hellobuddy
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20 :)
JamesJ
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what about for 180 degrees?
hellobuddy
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huh?
JamesJ
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That would correspond to a regular polygon with an infinite number of sides. What does that look like?
hellobuddy
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idk waht your asking
JamesJ
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It would look like a circle.
JamesJ
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As you add more and more sides to a regular polygon, it looks more and more like a circle. In the limit, as the number of sides goes to infinity, it would become a circle.
http://www.ck12.org/ck12/images?id=301055
hellobuddy
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thanks :)
hellobuddy
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g2g
JamesJ
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Here's an even better picture of this idea:
hellobuddy
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thanks!
cshalvey
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Great job JamesJ - and way to hang in there hellobuddy!