Assume for contradiction that a quadrilateral ABCD has
four internal obtuse angles, that is

m∠A > 90°
m∠B > 90°
m∠C > 90°
m∠D > 90°
---------

adding unequals to unequals in the same order:

m∠A + m∠B + m∠C + m∠D > 360°


  But the sum of the measures of the internal 
angles of an n-sided polygon is given by the
expression (n-2)180°.


So the sum of the measures of the internal 
angles of a quadrilateral, which is a 4-sided 
polygon, is given by the expression 
(4-2)180° = (2)180° = 360°.

Therefore

m∠A + m∠B + m∠C + m∠D = 360°

which contradicts the assumption that

m∠A + m∠B + m∠C + m∠D > 360°

Therefore the assumption is false.  Therefore
a quadrilateral cannot have 4 obtuse internal
angles.

Edwin