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hellobuddy

  • 2 years ago

if each interior angle of a regular polygon measures 144 degrees, how many sides does the polygon have?

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  1. hellobuddy
    • 2 years ago
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    Please show steps on how to solve.

  2. jesusisrisen
    • 2 years ago
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    obtuse angles but idont know about a polygon with obtuse angles

  3. JamesJ
    • 2 years ago
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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?

  4. hellobuddy
    • 2 years ago
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    the answer is supposed to be 10 sides how do you solve that?

  5. jesusisrisen
    • 2 years ago
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    decagon?

  6. JamesJ
    • 2 years ago
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    We'll get there ... follow me in logic first

  7. hellobuddy
    • 2 years ago
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    ok

  8. JamesJ
    • 2 years ago
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    What's the sum of angles of a square?

  9. hellobuddy
    • 2 years ago
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    180 (4-2) = 360

  10. Safiah
    • 2 years ago
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    360 ofcourse.as all angles are 90

  11. Safiah
    • 2 years ago
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    the polygon in question has 10 sides

  12. hellobuddy
    • 2 years ago
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    ^ why?

  13. JamesJ
    • 2 years ago
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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|

  14. hellobuddy
    • 2 years ago
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    i know how to do that, how do you solve when you dont know the number of sides?

  15. JamesJ
    • 2 years ago
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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 \[ \frac{180(n-2)}{n} \]

  16. JamesJ
    • 2 years ago
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    For your problem, set that equal to 144 and solve for n.

  17. Safiah
    • 2 years ago
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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

  18. hellobuddy
    • 2 years ago
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    so, James J, i do 180 (n-2) / n = 144 144-180+2= n/n?

  19. hellobuddy
    • 2 years ago
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    NOPE THATS NOT RIGHT ^

  20. hellobuddy
    • 2 years ago
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    i dont understnad..

  21. JamesJ
    • 2 years ago
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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?

  22. hellobuddy
    • 2 years ago
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    OH WOW I GET IT!

  23. hellobuddy
    • 2 years ago
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    sooo 180( n-2) = 144n wiat no, what do i do now?

  24. JamesJ
    • 2 years ago
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    ...hence \[ 180n - 360 = 144n \] Now?

  25. Safiah
    • 2 years ago
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    expand the brackets hellobudy

  26. hellobuddy
    • 2 years ago
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    nope.. now what?

  27. JamesJ
    • 2 years ago
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    Subtract 144n from both sides... \[ 180n - 144n - 360 = 0 \] i.e. \[ 36n - 360 = 0 \] Now can you finish it?

  28. hellobuddy
    • 2 years ago
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    -360 = -36 and then you divide 360 by 36 and you get ten

  29. JamesJ
    • 2 years ago
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    right, \[ 36n = 360 \] hence \[ n = \frac{360}{36} = 10 \]

  30. hellobuddy
    • 2 years ago
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    k i think i get it..

  31. hellobuddy
    • 2 years ago
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    thank you!

  32. JamesJ
    • 2 years ago
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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.

  33. hellobuddy
    • 2 years ago
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    ok! can you give me another example so i can practice? use a different degrees?

  34. hellobuddy
    • 2 years ago
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    ok ill try with 100 degrees

  35. JamesJ
    • 2 years ago
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    No, that won't work. One sec.

  36. JamesJ
    • 2 years ago
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    try 108 degrees

  37. hellobuddy
    • 2 years ago
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    ALRIGHT :)

  38. hellobuddy
    • 2 years ago
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    5 sides?

  39. JamesJ
    • 2 years ago
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    yes

  40. hellobuddy
    • 2 years ago
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    :D

  41. JamesJ
    • 2 years ago
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    last one, 162 degrees

  42. hellobuddy
    • 2 years ago
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    20 :)

  43. JamesJ
    • 2 years ago
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    what about for 180 degrees?

  44. hellobuddy
    • 2 years ago
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    huh?

  45. JamesJ
    • 2 years ago
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    That would correspond to a regular polygon with an infinite number of sides. What does that look like?

  46. hellobuddy
    • 2 years ago
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    idk waht your asking

  47. JamesJ
    • 2 years ago
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    It would look like a circle.

  48. JamesJ
    • 2 years ago
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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

  49. hellobuddy
    • 2 years ago
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    thanks :)

  50. hellobuddy
    • 2 years ago
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    g2g

  51. JamesJ
    • 2 years ago
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    Here's an even better picture of this idea:

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  52. hellobuddy
    • 2 years ago
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    thanks!

  53. cshalvey
    • 2 years ago
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    Great job JamesJ - and way to hang in there hellobuddy!

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