get Bent

[BMAD]

A bent is the union of the interior and boundary of a simple quadrilateral such that an interior angle formed by two adjacent edges exceeds 180◦ and the interior angle formed by the other two edges exceeds 90◦. Prove that the union of the interior and boundary of an acute triangle can not be the union of a finite number of bents which have disjoint interiors.

[k-man]

Take angle A of the acute triangle and evaluate what could be aligned with the side AB. There is only one way to position a "bent" inside an acute angle and align it with a side (see red bent in the picture). The angle C created by that "bent" and the continuation of triangle side AB will be acute. Attempting to position another "bent" into angle C whose side is aligned with AB will again create an acute angle D. Since there is no other way to align bents with AB there is no finite number of "bents" that can fill the side AB.

[bonanova]

I'm of two minds whether an infinite number of bents will cover it. You'd have to look at the area between the 180+ angle and a straight line. Can you take that limit and still have a bent?

[bonanova]

Take angle A of the acute triangle and evaluate what could be aligned with the side AB. There is only one way to position a "bent" inside an acute angle and align it with a side (see red bent in the picture). The angle C created by that "bent" and the continuation of triangle side AB will be acute. Attempting to position another "bent" into angle C whose side is aligned with AB will again create an acute angle D. Since there is no other way to align bents with AB there is no finite number of "bents" that can fill the side AB.

Bent.png

 

I'm thinking of how to fill up the entire angle A in k-man's drawing.

The reflex interior angle would have to approach 180 degrees to  do that.

That demands an infinite number of bents, whose angles approach 180 degrees as a limit.

 

The requirement of an infinite number of bents contradicts the terms of the OP.

Q. E. D.