Sorry that I cannot post a picture (I don't have 10 rep), so this might be confusing. Basically, I had a bunch of lines, two parallel, and 2 transverse lines going through them, making a triangle. The triangle only has an angle and 2 sides. However, since two of the lines are parallel, I am able to figure out that one of the angles of the triangle is 40 degrees, and another is 70 degrees.
Problem Now:
Lets say I now have $\triangle {ABC}$
- $\angle A = 40 ^\circ $
- $AB = 10 $
- $\angle B = 70 ^\circ $
- $BC = 7$
- $\angle C = 70 ^\circ $
- ** Find CA **
Using my head, I know the answer for CA is 10, since this is an isosceles triangle. However, what I don't get is why Law of Sines doesn't work here. What I did was set $\frac{\sin 40}{7} = \frac {sin(70)}{x}$. Solving for $x$, I got 10.23. Is it supposed to do this, or did I just mess up somewhere?
Your calculations are correct. If the question were: Is this triangle possible in flat Euclidean space? your answer would be No and your proof would be based on the law of sines.
Now, for fun, try to create a triangle with segments in this proportion. What happens?