Solve the geometry problem shown in the image.

Let's analyze the triangle shown in the image. We are given an isosceles triangle ABC, where AB = BC. We are also given that the exterior angle at vertex C is 125 degrees. Let's find the measure of angle B.

1. **Find the interior angle at C:** Since the exterior angle at C is 125 degrees, the interior angle \(\angle ACB\) can be found using the fact that the sum of an interior angle and its corresponding exterior angle is 180 degrees. Therefore,

\(\angle ACB = 180^\circ - 125^\circ = 55^\circ\)

2. **Find the angle at A:** Since triangle ABC is isosceles with AB = BC, the angles opposite these sides are equal. Thus, \(\angle BAC = \angle ACB = 55^\circ\).

3. **Find the angle at B:** The sum of the angles in a triangle is 180 degrees. Therefore,

\(\angle ABC = 180^\circ - (\angle BAC + \angle ACB) = 180^\circ - (55^\circ + 55^\circ) = 180^\circ - 110^\circ = 70^\circ\)

So, the measure of angle B is 70 degrees.

**Final Answer:** The measure of angle B is 70 degrees.