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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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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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