To begin, we use the property of supplemental angles as the angle inside the smaller triangle (to the right of the 65 degree angle) is 115 degrees. That lets us look at the far right triangle, which has an angle, side, and angle in that order. This ASA triangle can be solved by the Law of Sines.
What we can do immediately is find one of the top angles. Since the sum of the angles in the triangle is 180, then the other angle of the obtuse triangle is 26 degrees, from 180 - 115 - 39. We now know all 3 angles of the triangle that has 37 feet as one of its sides. We want to find the side opposite the 39 degree angle. That side is in common in both triangles and makes up the triangle with the tower, too.
Let x be the unknown side. We use the Law of Sines, which tells us to put the side and its opposite angle in proportion with another side-opposite angle pair. By the Law of Sines,
x / sin (39°) = 37 / sin (28°)
37 sin 39° = x * sin 26°
37 sin 39° / sin 26° = x
23.28485 / .43837 = x
53.11676 = x
We have found the common side in both triangles. Now we move to the next triangle. Typically, towers stick straight up so the tower and the ground are perpendicular. That makes that angle 90 degrees, and the other angle in that triangle is 35 degrees (180 - 95 - 65). Again, we use the Law of Sines. Let y be the height of the tower.
y / sin 65° = 53.11676 / sin 90°
As the sine of 90 degrees is 1, we can simplify the proportion to this:
y / sin 65° = 53.11676
y = 53.11676 / sin 65° by dividing the sine of 65 degrees on both sides (remember, its a number)
y = 53.11676 / .90631
y = 58.6078
Thus, the tower is about 58 feet high and choice D represents the best answer.
So the tower is about 55 feet high. While it doesn't match our choices, we've used rounding
