Two triangles can be formed with the given information. use the law of sines to solve the triangles. a = 61°, a = 17, b = 19 b = 77.8°, c = 41.2°, c = 12.8; b = 102.2°, c = 16.8°, c = 5.6 b = 12.2°, c = 106.8°, c = 18.6; b = 167.8°, c = 73.2°, c = 18.6 b = 77.8°, c = 41.2°, c = 22.6; b = 102.2°, c = 16.8°, c = 22.6 b = 12.2°, c = 106.8°, c = 15.5; b = 167.8°, c = 73.2°, c = 15.5

The Law of Sines applies to any triangle and works as follows:

a/sinA = b/sinB = c/sinC

We are attempting to solve for every angle and every side of the triangle. With the given information, A = 61°, a = 17, b = 19, we can solve for the unknown angle that is B.

a/sinA = b/sinB
17/sin61 = 19/sinB
sinB = (19/17)(sin61)
sinB = 0.9774
sin-1(sinB) = sin-1(0.9774)
B = 77.8°

With angle B we can solve for angle C and then side c.

A + B + C = 180°
C = 180° - A - B
C = 180° - 61° - 77.8°
C = 41.2°

a/sinA = c/sinC
17/sin61 = c/sin41.2
c = 17(sin41.2/sin61)
c = 12.8

The first solved triangle is:

A = 61°, a = 17, B = 77.8°, b = 19, C = 41.2°, c = 12.8

However, when we solved for angle B initially, that was not the only possible answer because of the fact that sinB = sin(180-B).

The other angle is simply 180°-77.8° = 102.2°. Therefore, angle B can also be 102.2° which will give us different values for c and C.

C = 180° - A - B
C = 180° - 61° - 102.2°
C = 16.8°

a/sinA = c/sinC
17/sin61 = c/sin16.8
c = 17(sin16.8/sin61)
c = 5.6

The complete second triangle has the following dimensions:

A = 61°, a = 17, B = 102.2°, b = 19, C = 16.8°, c = 5.6

The answer you are looking for is the first option given in the question:

B = 77.8°, C = 41.2°, c = 12.8; B = 102.2°, C = 16.8°, c = 5.6

Using the rise/run rule. you would get 2/6.simp. 1/3