Law of Sines and alternate interior angles?

[benign.psycho]

Homework Statement

A ranger in tower A spots a fire at a direction of 321 degrees. A ranger in tower B, located 60 mi at a direction of 47 degrees from tower A, spots the fire at a direction of 279 degrees. How far from tower A is the fire? How far from tower B?

Homework Equations

http://img19.imageshack.us/img19/9529/triangle.jpg

The Attempt at a Solution

Angle A is 86 degrees. From here, I'm not exactly sure how to get the solution. I know that angle C is given from (321 degrees - 279 degrees), which is 42 degrees. Once I have angle C, I can solve for everything else.

I know this has something to do with alternate interior angles, but can't quite grasp why the subtraction is taking place. On another problem similar to this, I'm even more confused because the subtraction yields nothing of importance.

Can you guys explain this clearly for me, any rules or laws that are used here?

Thanks!

 

[benign.psycho]

Nevermind, figured it out, feel stupid.

Heh.

 

[Architeuthis Dux]

I can explain the concept of alternate interior angles and how it relates to the given problem. The law of sines states that in a triangle, the ratio of a side length to the sine of its opposite angle is constant. In this case, we have two triangles formed by the two rangers and the fire, with the distance between the two towers as the common side. By using the law of sines, we can set up the following equations:

For triangle A:

sin(86 degrees) / x = sin(42 degrees) / (60 - x)

For triangle B:

sin(86 degrees) / y = sin(42 degrees) / (60 + y)

Where x is the distance from tower A to the fire, and y is the distance from tower B to the fire.

Now, to understand why we are subtracting the angles, we need to look at the angles formed by the two towers and the fire. These angles are alternate interior angles, meaning they are on opposite sides of the transversal (the line connecting the two towers) and are formed by two parallel lines (the line of sight from each tower to the fire and the line connecting the two towers). According to the alternate interior angle theorem, alternate interior angles are congruent, which means they have the same measure. So, by subtracting the angle formed by tower B (279 degrees) from the angle formed by tower A (321 degrees), we get the angle formed by the fire (42 degrees).

By solving the two equations above, we can find the values of x and y, which will give us the distances from each tower to the fire. I hope this explanation helps you understand the concept of alternate interior angles and how it is used in this problem.