Line segment AB has a length of 15 and angle a=35degrees . A segment with a length of 12 will form the third side of the triangle. What are the possible measures of the angle opposite side AB? Please explain the process.

Only SSA gives the possibility of 2 answers. This one does not give that opportunity. There is one unique answer. We'll discuss 2 and zero after finding 1 answer. On looking at it again, the question might be ambiguous. We'll check that out as well.

Given

AB = 15

<A = 35

point C opposite line AB such that CB = 12 These givens give a unique answer.

Solution

Sin(35) / 12 = Sin(C) / 15 Multiply both sides by 15

15*Sin(35) / 12 = Sin(C) Find 15/12

1.25*sin(35) = Sin(C) Write Sin(35)

1.25*0.5736 = Sin(C) Multiply the left

0.71697 = Sin(C) Take the inverse Sin

C = 45.805 degrees This is the angle opposite AB

Angle B = 180 - 35 - 45.805 = 99.2

Ambiguous Case

If AC = 12 we have another answer entirely. This is SAS which will give just 1 set of answers for the triangle. The reason the case is ambiguous is because we don't exactly know where that 12 unit line is. It could be AC or BC.

I will set up the Sin law for you, and let you solve it

Sin(B) / 12 = Sin(35)/15

When you solve for Sin(B) as done above you, get 0.45886 from which B = 27.31 degrees

C = 180 - 35 - 27.31 = 117.69

So that's two values that C could have. I think that's all given these conditions.

Two Cases or None

<A = 35 degrees

AC = 15

CB = 12

This should give you two possible cases or none. You can check which by finding the height of the triangle from C down to AB (which has no distinct length. The h is 15 * Sin35 = 8.6. If CB < 8.6, there are no solutions. If CB < AC then if CB > that 8.6, there are 2 solutions.