Angles of convex polygons












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For a convex polygon, show that the sum of any two interior angles is greater than the difference between any two interior angles. (the polygon has more than 3 sides)



If I pick 4 dots A,B,C,D and say the theorem applies(angleA+ angleB>angleC-angleD), then it is same as proving that angleA+angleB+angleD>angleC, so I found that this theorem is the same meaning by saying that sum of any three interior angles is greater than any one angle. However, since this is a convex polygon, one Angle is bigger than zero and smaller than one hundred eighty. Then I want to prove that the sum of three angles in a convex polygon is bigger than 180(or equal). Is there any proof that works for every convex polygon?










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    0












    $begingroup$


    For a convex polygon, show that the sum of any two interior angles is greater than the difference between any two interior angles. (the polygon has more than 3 sides)



    If I pick 4 dots A,B,C,D and say the theorem applies(angleA+ angleB>angleC-angleD), then it is same as proving that angleA+angleB+angleD>angleC, so I found that this theorem is the same meaning by saying that sum of any three interior angles is greater than any one angle. However, since this is a convex polygon, one Angle is bigger than zero and smaller than one hundred eighty. Then I want to prove that the sum of three angles in a convex polygon is bigger than 180(or equal). Is there any proof that works for every convex polygon?










    share|cite|improve this question











    $endgroup$















      0












      0








      0





      $begingroup$


      For a convex polygon, show that the sum of any two interior angles is greater than the difference between any two interior angles. (the polygon has more than 3 sides)



      If I pick 4 dots A,B,C,D and say the theorem applies(angleA+ angleB>angleC-angleD), then it is same as proving that angleA+angleB+angleD>angleC, so I found that this theorem is the same meaning by saying that sum of any three interior angles is greater than any one angle. However, since this is a convex polygon, one Angle is bigger than zero and smaller than one hundred eighty. Then I want to prove that the sum of three angles in a convex polygon is bigger than 180(or equal). Is there any proof that works for every convex polygon?










      share|cite|improve this question











      $endgroup$




      For a convex polygon, show that the sum of any two interior angles is greater than the difference between any two interior angles. (the polygon has more than 3 sides)



      If I pick 4 dots A,B,C,D and say the theorem applies(angleA+ angleB>angleC-angleD), then it is same as proving that angleA+angleB+angleD>angleC, so I found that this theorem is the same meaning by saying that sum of any three interior angles is greater than any one angle. However, since this is a convex polygon, one Angle is bigger than zero and smaller than one hundred eighty. Then I want to prove that the sum of three angles in a convex polygon is bigger than 180(or equal). Is there any proof that works for every convex polygon?







      angle convex-geometry






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      edited Dec 28 '18 at 6:03







      mathhero

















      asked Dec 28 '18 at 5:12









      mathheromathhero

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      235






















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          $begingroup$

          It's not true. Try in triangle $alpha=120^{circ}$ and $beta=gamma=30^{circ}.$



          Thus, $$beta+gamma>alpha-beta$$ is wrong.






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            $begingroup$

            It's not true. Try in triangle $alpha=120^{circ}$ and $beta=gamma=30^{circ}.$



            Thus, $$beta+gamma>alpha-beta$$ is wrong.






            share|cite|improve this answer











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              0












              $begingroup$

              It's not true. Try in triangle $alpha=120^{circ}$ and $beta=gamma=30^{circ}.$



              Thus, $$beta+gamma>alpha-beta$$ is wrong.






              share|cite|improve this answer











              $endgroup$
















                0












                0








                0





                $begingroup$

                It's not true. Try in triangle $alpha=120^{circ}$ and $beta=gamma=30^{circ}.$



                Thus, $$beta+gamma>alpha-beta$$ is wrong.






                share|cite|improve this answer











                $endgroup$



                It's not true. Try in triangle $alpha=120^{circ}$ and $beta=gamma=30^{circ}.$



                Thus, $$beta+gamma>alpha-beta$$ is wrong.







                share|cite|improve this answer














                share|cite|improve this answer



                share|cite|improve this answer








                edited Dec 28 '18 at 5:37

























                answered Dec 28 '18 at 5:30









                Michael RozenbergMichael Rozenberg

                108k1895200




                108k1895200






























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