In an isosceles triangle, if one pair of base angles is twice the measure of the second pair of base angles, what are the measures of the angles?

A triangle has 3 angles. Here you are mentioning two pairs of angles, which is four angles.

Then I believe that the figure may be an isosceles trapezoid, instead of a triangle.

The trapezoid is a quadrilateral.

In an isosceles trapezoid there are two pair of equal base angles. Call the angles x, x, y and y.

The internal angles of a quadrilateral sum 360 degrees, then 2x + 2y = 360.

Given that the measure of one pair is twice the measure of the other, you can  state that 2y = 4x and 2x + 4x = 360

Now solve the equation:

6x = 360

x = 360 / 6

x = 60.

Then the measures are 60, 60, 120 and 120 degrees.

 

Let the other angle of the isosceles triangle be x, then
2x + 2x + x = 180
5x = 180
x = 180/5 = 36

Therefore, the measures of the triangle is 36°, 72°, 72°