Explain why the decagons angles equal 1440 degrees?

Answer 1

Here is one way to prove it. It is easier if the decagon is convex but the proof is valid even if not.

Take any point P in the plane. Again, it is easier if this point is inside the decagon but the proof is valid even if it is not.

Join all the vertices of the decagon to this point. So now you have ten triangles with a common vertex P.

The angles of these ten triangles form the internal angles of the decagon plus all the angles around P.

So the sum of the internal angles of the decagon is equal to the sum of the internal angles of all ten triangles with P as their common vertex minus the 360 degrees worth of angles at P itself.

The sum of the interior angles of each triangle is 180 degrees so the sum of ten triangles with common vertex P is 180*10 = 1800 degrees.

So the internal angles of the decagon add to 1800 - 360 = 1440 degrees.