The measure of angle $A'B'C'$ is 60 degrees
Triangles $ABC$ and $A'B'C'$ are similar because they have two equal angles: angle $B$ and angle $B'$. This is because sides $AB$ and $A'B'$ are parallel, and alternate interior angles are equal when parallel lines are cut by a transversal.
Since the triangles are similar, their corresponding angles are also equal. Therefore, angle $A'B'C'$ is equal to angle $A$.
We know that the sum of the angles in a triangle is 180 degrees. In triangle $ABC$, the sum of angles $A$ and $B$ is 100 degrees (40 degrees + 60 degrees). Therefore, angle $C$ must measure 80 degrees (180 degrees - 100 degrees).
This means that angle $A$ must measure 100 degrees - 80 degrees = 20 degrees.
Additionally, here is a possible mathematical solution:
Let $x$ be the measure of angle $A'B'C'$. We know that the sum of the angles in a triangle is 180 degrees, so we can write the following equation:
x + 40 degrees + 60 degrees = 180 degrees
Combining like terms, we get:
x + 100 degrees = 180 degrees
Subtracting 100 degrees from both sides, we get:
x = 180 degrees - 100 degrees