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The problem is asking for two possible sets of angle measures for an isosceles triangle, given that one of its exterior angles is 150°. Let's walk through the solution.

### Key points:
1. **Exterior angle property**: The exterior angle of a triangle is equal to the sum of the two interior opposite angles. So, for an exterior angle of 150°, the sum of the two opposite interior angles is:
   \[
   180^\circ - 150^\circ = 30^\circ
   \]
   These two opposite interior angles must be equal because the triangle is isosceles, so each base angle will be:
   \[
   \frac{30^\circ}{2} = 15^\circ
   \]
2. **Sum of angles in a triangle**: The sum of all interior angles of a triangle is always \(180^\circ\). Since each base angle is \(15^\circ\), the vertex angle \(V\) can be calculated as:
   \[
   V = 180^\circ - 15^\circ - 15^\circ = 150^\circ
   \]

Thus, **one possible set of angles** is:
- **Base angles**: \(15^\circ\)
- **Vertex angle**: \(150^\circ\)

### Second Possible Set:
If the exterior angle is 150° and we interpreted the base angles as equal, there is a special case where **the triangle could still be isosceles but different proportions**. However, since the exterior angle is associated with only one case here (since the sum of two interior angles is uniquely determined), there is no second distinct solution in the general case for an isosceles triangle.

The two angles provided (15° and 150°) seem to reflect the correct base angle and vertex angle measures.

Would you like further clarifications?

### Related Questions:
1. What are the properties of an isosceles triangle's angles?
2. How do exterior angles relate to the interior angles of a triangle?
3. Why must the sum of a triangle's interior angles always equal 180°?
4. Can an isosceles triangle have more than one solution for its angle measures?
5. How can we generalize the relationship between exterior angles and interior angles for any triangle?

### Tip:
Always remember that for any polygon, the exterior angles will always sum to 360°, no matter the number of sides!

An exterior angle of an isosceles triangle has measure 150°. Find two possible sets of measures for the angles of the triangle. If the exterior angle of the base is 150°, what are the measures of each base angle and the vertex angle?

This problem involves calculating the interior angles of an isosceles triangle with an exterior angle of 150°. The solution includes key concepts like the exterior angle theorem and the sum of angles in a triangle. Suitable for students in Grades 8-10.

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