The problem asks for the linear angle of the dihedral angle with edge BC, given that $PA \perp (ABC)$ and $\angle C$ in triangle $ABC$ is $90^\circ$.

Here's how we approach the problem:

1. Perpendicularity of PA to Plane ABC: Since $PA \perp (ABC)$, the line $PA$ is perpendicular to the plane containing triangle $ABC$. This fact helps us determine that angles formed between $PA$ and lines in the plane (such as $BC$) involve considering right angles.

2. Right Triangle ABC: In triangle $ABC$, we know that $\angle ACB = 90^\circ$. This implies that line $BC$ is the hypotenuse of right triangle $ABC$.

3. Dihedral Angle with Edge BC: The question asks for the linear angle associated with the dihedral angle between two planes, one of which has the edge BC as its common line. The other plane involved should likely be formed by the line $PA$, which is perpendicular to the plane $ABC$.

4. Angle Calculation: Since $PA$ is perpendicular to $BC$, the linear angle of the dihedral angle between the planes should involve the triangle $ACP$, where $\angle ACP$ would be relevant.

From the given choices, the correct answer would be ACP, as it captures the angle formed by the plane containing $PA$ and the plane containing the triangle $ABC$, considering the edge $BC$.

Would you like further clarification on any step?

Here are 5 related questions:

1. What is the geometric significance of a dihedral angle?
2. How do you find the linear angle of a dihedral angle in general?
3. How does the condition $PA \perp (ABC)$ simplify the problem?
4. What are the key properties of right triangles that apply here?
5. How do perpendicular planes affect the angles between intersecting lines?

Tip: When dealing with dihedral angles, visualize the planes involved and their common edge to find the corresponding linear angle.