To determine the correct option, we need to understand the relationship between the angles and the sides in triangle $\triangle ABC$.

In any triangle, the side opposite the largest angle is the longest, and the side opposite the smallest angle is the shortest. Given:

• $\angle A = 45^\circ$
• $\angle B = 60^\circ$
• $\angle C = 75^\circ$

Since $\angle C$ is the largest angle (75°), the side opposite it, which is $\overline{AB}$, will be the longest side. Similarly, since $\angle A$ is the smallest angle (45°), the side opposite it, which is $\overline{BC}$, will be the shortest side. This gives us the following side length order:

$\overline{BC} < \overline{AC} < \overline{AB}$

The correct option is the first choice:

$\overline{BC} < \overline{AC} < \overline{AB}$

Would you like further details or have any questions on this?

Here are some related questions you might find helpful:

1. How do we determine side lengths based on angle measures in any triangle?
2. What is the Triangle Inequality Theorem, and how does it apply here?
3. How can we use trigonometry to find the actual side lengths in a triangle if angles are known?
4. What are the different types of triangles based on their angle measures?
5. Why does the side opposite the largest angle have to be the longest?

Tip: Remember, in any triangle, larger angles are always opposite longer sides.