Why does all quadrilaterals add up to 360 degrees?

Answer 1

Proof:Draw any quadrilateral. Draw a line from one corner to the opposite corner. This line is called a diagonal and quadrilaterals have two of them. Some quadrilaterals won't have both diagonals completely within the quadrilateral but they all have at least one diagonal with this property. Using this diagonal it is clear to see the quadrilateral is divided into 2 triangles and that adding up all the angles in the triangle will give you the sum of the quadrilateral. Since the angle sum of any triangle is 180 degrees and we have two triangles we have 360 degrees.
Generalisation:Any polygon with n corners (called vertices) has n edges. You can take a polygon with n-1 vertices for n>=4 and stick a triangle onto one edge and you increase the angle sum by 180 degrees. So the angle sum of a polygon with n vertices is 180*(n-2) for n>=3.