what is the maximum number of acute angles

Question

What is the maximum number of acute angles in a convex 10-gon in the Euclidean plane ?

I know that the answer is at least $4$.

Any idea how to proceed.

Answer 1

No, the answer is $3$, in fact no convex polygon can have more than $3$ acute angles, this is because the sum of the external angles is $360$, and an external angle at a vertex for an internal acute angle is larger than $90$ degrees.

Answer 2

The angles of the 10-gon need to sum up to $8\cdot 180^\circ = 1440^\circ$. If there are $4$ acute angles, then they sum up to less than $360^\circ$, which means the remaining $6$ angles need to sum up to more than $1080^\circ$, impossible since each one is at most $180^\circ$.

On the other hand, to have $3$ acute angles, draw four consecutive segments that almost form a square, and connect the two end vertices with six short segments.

Answer 3

You're going to have a tough time making it 4. Think about this: If you have an obtuse angle in the 10-gon, replace it with a straght line. You remove one angle and the two angles that you modified got more acute. Now prove that the largest $n$ such that you have a convex $n$-gon with only acute angles is 3.