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Can a non-convex polyhedral angle have the sum of plane angles smaller than $180^\circ$?


I don't have a fully-fledged attempt. I've only been able to gather some potentially useful facts.

Here's what I got:

Given: non-convex polyhedral angle SABCD, where sides ASD and DSC are concave.

  • $|ASD-DSC| < ASC < ASD+DSC$
  • $ASD+DSC+CSB+BSA<360$
  • |difference of two plane angles adjacent angles| < angle < sum of two adjacent angles

I'm sure the answer is "no", out of intuition. How can I prove it, though?

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  • $\begingroup$ What do you mean by non convex polyhedral angle? $\endgroup$ – Del Oct 18 '16 at 20:52
  • $\begingroup$ @Del -- According the the textbook from which this question is posed (Kiselev's Geometry, Book 2: Stereometry), "non-convex" means concave. $\endgroup$ – Fine Man Oct 19 '16 at 18:09

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