Question 466114
<pre>
Assume for contradiction that a quadrilateral ABCD has
four internal obtuse angles, that is

m&#8736;A > 90°
m&#8736;B > 90°
m&#8736;C > 90°
m&#8736;D > 90°
---------

adding unequals to unequals in the same order:

m&#8736;A + m&#8736;B + m&#8736;C + m&#8736;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&#8736;A + m&#8736;B + m&#8736;C + m&#8736;D = 360°

which contradicts the assumption that

m&#8736;A + m&#8736;B + m&#8736;C + m&#8736;D > 360°

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

Edwin</pre>