In the image, triangle $ABC$ is dilated with a scale factor of $\frac{1}{3}$, and we are tasked with labeling the angle measurements in triangle $A'B'C'$.

Since dilation preserves the angle measurements of a triangle, the angles in triangle $A'B'C'$ will be identical to those in triangle $ABC$. Here’s how to label the angles in $A'B'C'$ based on the provided measurements in triangle $ABC$:

1. $\angle A'$ corresponds to $\angle A = 21^\circ$.
2. $\angle B'$ corresponds to $\angle B = 51^\circ$.
3. $\angle C'$ corresponds to $\angle C = 108^\circ$.

Thus, the angle labels for triangle $A'B'C'$ should be:

• $\angle A' = 21^\circ$
• $\angle B' = 51^\circ$
• $\angle C' = 108^\circ$

Would you like further details on the dilation process or more information on similar triangles?

Here are some additional questions you might find interesting:

1. What happens to the side lengths when a triangle is dilated with a scale factor of $\frac{1}{3}$?
2. How does a scale factor affect the perimeter of a dilated triangle?
3. Does dilation affect the area of a triangle? If so, by what factor?
4. What are the properties of similar triangles?
5. How would the angle measures change if the dilation was a reflection instead?

Tip: When working with dilations, remember that angle measures stay the same, but side lengths are scaled by the given factor.