Final answer:
The measure of ∠ABC is 120 degrees, determined by utilizing the sum of angles in a triangle and the properties of an equilateral triangle.
Step-by-step explanation:
To determine the measure of ∠ABC when an equilateral triangle and an isosceles triangle share a common side, we must first recall that the sum of the angles in any triangle is 180 degrees. In an equilateral triangle, all sides, as well as all angles, are equal, meaning each angle is 60 degrees. Now, let's assume the equilateral triangle is ∆ABC and the isosceles triangle shares side AB with ∆ABC.
Since ∆ABC is equilateral, we know that ∠A and ∠B each measure 60 degrees. If we label the vertices of the isosceles triangle as A, B, and D, where BD is the unequal side, ∠ABD (or ∠ABC since they share the same vertex B in this configuration) can be found by subtracting ∠A from the sum of angles in ∆ABD, which is 180 degrees. This leaves us with ∠ABC = 180 degrees - ∠A = 180 degrees - 60 degrees = 120 degrees. Therefore, ∠ABC measures 120 degrees.