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

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?

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