Suppose you have triangle ABC with the given information.

The measure of angle A = 65
Meansure of angle B = 35
Length of side a = 12

What is the measure of angle C and the lengths of sides b and c?

If this was a right triangle, we could simply use sine, cosine of tangent to get side b or c and the other side would be obtained using the Pythagorean Theorem.

But since this is not a right triangle (measure of angle C = 80), we can use the Law Of Sines which states

SinA/a = SinB/b = SinC/c (alternatively it be can be written a/SinA = b/SinB = c/SinC)

We will use

SinA/a = SinB/b to get the length of side b

therefore

Sin(65)/12 = Sin(35)/b

12*Sin(35) = b*Sin(65)

6.88 = 0.906b

b = 7.6

Now we can get c using

SinA/a = SinC/c

Sin(65)/12 = Sin(80)/c

12*Sin(80) c*Sin(65)

11.82 = 0.906c

c = 13.05