Find the unknown angles in triangle ABC for each triangle that exists.

A = 142.13°, b = 5.432 ft, a = 7.297 ft

Video transcript

Welcome back. Everyone. In this problem, we want to determine the measure of the remaining angles of a triangle. ABC with the measurements that says the angle A is 151.69 degrees. The side B is 8.796 ft and the side A is 10.654 ft. For our answer choices A says that the angle B is 13.05 degrees and the angle C is 15.26 degrees. B says it's 5.26 degrees and 23.05 degrees. C says it's 23.05 degrees and 5.26 degrees respectively. And D says that no triangle exists. Now, before we determine the measure of the remaining angles, it would help to draw a sketch of what our triangle looks like. So let's draw our triangle here. OK. And it would probably look something like this where our angle A or obtuse angle is 151.69 degrees. Now, we know that for a triangle in its naming convention for the angle A, the opposite side is common letter A likewise if we label our angles B and C on our triangle here, then B would be opposite to angle B and side C would be opposite to angle C. And we already know that side A equals 10.654 ft. While side B, let me make some space here. Side B equals 8.796 ft. That's side B and now we are trying to figure out, oops, I have some, a little trouble going on here. Let me write that properly. And we're trying to figure out the value of sides B are angles B and angle C. Now let's think about what we know what we have here. So already we know that we have a pair of angles and sides, we have side A and it's the angle opposite to side A. We're trying to figure out the side B and we also know the side that is opposite to angle B. OK. So we're trying to figure out angle B and we know the side that's opposite to angle B. So do we know anything that relates opposite sides to their angles? Well, we know that we have the sign rule. OK. Or the law of science. And recall, let me put that in blue here to show that we're solving for angle B. Recall that by the law of science by the last signs, the love of science basically tells us that the ratio of sides to their opposite angles throughout a triangle are the same. So in other words, the ratio of the sine of angle A two side A equals the ratio of the sine of B to side B, which is the same as the sine of C to side C. So here, since we have the sine of A and the value of S A and we have the value of SB but we want to figure out the value of angle B, we can use this relationship to help us because no, by the sign rule, this tells us then that the S of B are unknown angle divided by the side B 8.796 ft would be equal to the sine of anger A which is 151.69 degrees two, it's opposite side A which is 10.654 ft. OK. What does that mean? Well, no, if we're going to find a sign of B, we can go ahead and solve. So let me come down here because this tells us then that the sine of B is going to be equal to 8.796 multiplied by the sine of 151.69 degrees all divided by 10.654. If we want to get B by itself, that means we can find the inverse sign of both sides. And that would tell us then that B is going to be equal to the inverse sign of or expression. And I haven't simplified our expression yet because I want us to put it directly in the calculator so that we can get the exact value. OK. So B would be equal to the inverse sine of 8.796 sine of 151.69 divided by 10.654. Now, we can put this expression into our calculator to solve for B recall that all of our answers were written rounded to two decimal places. So we'll do the same with ours. So when we put this into our calculator, put this expression, we should find that angle B is approximately equal to 23.05 degrees to two decimal places. So that's the value for angle B. Now, let's see if we can figure out the value for angle C. Let's move a bit to the right here. OK. Now, if we know that B is 23.05 degrees, and we already know that angle A is 151.69 degrees, we can use both of those angles to solve for angle C because we're dealing with a triangle here and recall that the sum of angles in a triangle equals 180 degrees. So let me put that in red by the UN on property of triangles, we know that an air 151.69 degrees plus B 23.05 degrees plus angle C is going to equal 180 degrees. Therefore, angle C must be equal to C must be equal to 180 degrees minus 151.69 degrees plus 23.05 degrees. And now again, if we try to solve for C by putting that into our calculator, we should find that C is equal to 5.26 degrees. Now, just to make sure that these are the only possible values for C and B, let's think about our possibilities here. There could also be another value for B that we could find by subtracting this value for B 23.05 degrees from 180. So let me put an asterisk here. The other value for B could be 180 degrees minus 23.05 degrees. And when we put that into our calculator, we'll get 156.95 degrees. And if this is the other value for B, then that means the sum of A and B would be equal to 151.69 degrees plus 156.95 degrees, which would equal 308.64 degrees. But here we've run into a problem A and B, the sum of A and B are greater than 180 degrees. And we haven't even added C yet. Therefore, 156.95 degrees could not be a valid measure for B because if we add it to the rest of angles in our triangle, the result would be greater than 180 degrees. Therefore, the values of B or, or, or the values of our missing angles are B equals 23.05 degrees and the C equals 5.26 degrees. If we look back on our answer choices, that would have been answer choice. C Thanks a lot for watching everyone. I hope this video helped.