The four interior angles of an n-side polygon is given by the formula

 = ×180°

A quadrilateral has 4 sides so,

 = (4-2)×180° = 2×180° = 360°

If we wish to divide a number N into 4 parts in the ratio of a:b:c:d, the
parts are given by these formulas:

×N, ×N, ×N, and ×N

So since we wish to divide 360° into 4 parts in the ratio of 2:4:5:7 


×360° = ×360° = ×360° =  = 40°

×360° = ×360° = ×360° =  = 80°

×360° = ×360° =  = 100°

×360° = ×360° =  = 140°

We can check by adding the 4 angles to see if we get 360°

                  40° + 80° + 100° + 140° = 360°

We get 360°, so we must not have made any mistakes.

Edwin