To solve this problem, let's break down the situation and apply the law of sines.

Step 1: Diagram and Label the Points

1. Let A be the location of tower A.
2. Let B be the location of tower B, which is 60 miles away from tower A in the direction of 40 degrees from A.
3. Let F be the location of the fire.

We know:

• The angle of sight from A to the fire, $\angle FAX = 324^\circ$.
• The angle of sight from B to the fire, $\angle FBX = 323^\circ$.
• The distance $AB = 60$ miles.

Step 2: Determine the Internal Angles of Triangle ABF

Convert these bearings into internal angles within the triangle $\triangle ABF$:

1. The direction from A to B is 40 degrees, and the fire is spotted from A at 324 degrees. This difference gives the angle at A: $\angle A = 324^\circ - 40^\circ = 284^\circ.$