Each face of a pyramid is an isosceles triangle with a 82 degree vertex angle. What are the measures of the base angles?
In an isosceles triangle, the base angles are congruent.
Let's call the base angles of the isosceles triangle $x$.
Since the vertex angle is 82 degrees, the sum of the base angles is equal to $180 - 82 = 98$ degrees.
Since the base angles are congruent, we can write an equation:
$2x = 98$
Now, divide both sides by 2:
$x = 98/2$
Simplifying:
$x = 49$
Therefore, each base angle of the isosceles triangle is $\boxed{49}$ degrees.