The angle opposite the base of an isosceles triangle is 50 degrees. Find the outside angle at the base.

The triangle sum theorem says that the sum of the angles of any triangle is 180 degrees.
In an isosceles triangle, the angles at the base are equal (angle 1 = angle 3 = x), therefore:
x + angle 2 + x = 180 degrees;
2x + 50 degrees = 180 degrees;
2x = 130 degrees;
x = 65 degrees.
Angle 1 = Angle 2 = 65 degrees.
The outer base angle (angle 4) and the inner base angle are adjacent, so their sum is 180 degrees:
angle 3 + angle 4 = 180 degrees;
65 degrees + angle 4 = 180 degrees;
angle 4 = 115 degrees.
Answer: 115 degrees.