Question

Decide whether the statement is always, sometimes, or never true.


The measure of an exterior angle of a triangle is greater than the sum of the measures of the two opposite interior angles.

Answer 1

Draw a simple triangle. Label the interior angles x, y, and z. Draw the exterior angle for the angle marked z, and call it z'. Now, consider what you know about a triangle: all interior angles must add up to 180 degrees. Therefore:
\[x+y+z=180\]Also, since the angle z' is exterior to angle z, z' and z must be supplementary (add up to 180 degrees):
\[z'+z=180\]Since both statements are true, solve the second one for z, substitute your results into the original equation, and see what the relationship is between x, y, and z'.

Answer 2

...so always sometimes or never? i have to solve this quick. =\

Answer 3

\[z=180-z'\]\[x+y+(180-z')=180\]\[x+y-z'=0\]\[x+y=z'\]\[z'=x+y\]In other words, the exterior angle is equal to the sum of the two interior opposite angles.