Geometry homework, i'll give brainliest

There are a few facts we'll need to hold onto for this problem:

- Supplementary angles (two angles that form a straight angle when added) add to 180°
- The interior angles of a triangle add to 180°
- Sides opposite congruent angles in a triangle are also congruent
- Alternate interior angles are congruent when formed by a transversal through two parallel lines

With that in mind, let's get to the problem. First, what data have we been given?

Let's drop another point on the ground to the left of point T called S, so we can get all of this data together. We know that:

 

We don't know the measure of ∠ATB, but we do know that it shares the larger angle ∠BTS (which is 80°) with ∠ATS (which is 40°). Since ∠ATB + ∠ATS = ∠BTS, we can clearly see that ∠ATB must also be 40°, so now we have the new data that ∠ATB = 40° to work with.

Next, notice that ∠BTS and ∠CBT are alternate interior angles, and the lines they rest on,  and , are parallel. This implies that ∠BTS and ∠CBT are congruent, which makes ∠CBT = 80°. 

∠CBT and its neighboring angle ∠ABT, when combined, form a straight angle, making them supplementary to one another. This means that ∠CBT + ∠ABT = 180°, or 80° + ∠ABT = 180°. Subtracting 80° from both sides, we see that ∠ABT = 100°.

We're almost done; we now bring our attention now to the triangle formed by the points ABT. What do we know about this triangle? We're given that one of its sides, AB, is equal to 2.4 mi; most recently, we determined that one of its angles, ∠ABT, was equal to 100°; and even earlier, we found that the second one of its angles, ∠ATB, was equal to 40°. We have two angles and a side, which, as I'm about to demonstrate, is more than enough for our purposes. If we've found the measures of two angles of triangle ABT, we can use the fact that a triangle's interior angles add to 180° to find the third angle, ∠BAT. We have:

∠ABT + ∠BTA + ∠BAT = 180°

Substituting ∠ABT and ∠BTA for the values we found for them:

100° + 40° + ∠BAT = 180°
140° + ∠BAT = 180°
∠BAT = 40°

We now know that ∠BAT and ∠BTA are congruent, which means that the triangle BTA is isoceles. This means that the sides opposite those angles will also be congruent. The side opposite ∠BTA is the line segment AB, which we know to be 2.4 mi, which means the side opposite ∠BAT, BT - the side we've been looking for - must also be 2.4 mi.