Question

Determine the number of possible triangles, ABC, that can be formed given B = 45°, b = 4, and c = 5. 0

Answer 1

We have that

m∠B = 45°  b = 4  c = 5

we know that

from the Law of Sines, obtain
sin C/c=sin B/b
sin C=c*sin B/b-------> 5*sin 45/4---------> 0.884
C=arc sin(0.884)---------> C=62.1°
A+B+C=180°
so
m∠A = 180 - 45 - 62.1 = 72.9°
a=(sin A/sin B)*b--------> (sin 72.9/sin 45)*4------> 5.41

The first triangle has∠A=72.9°,  m∠B=45°,  m∠C = 62.1°,  a=5.41,  b=4,  c=5.

Also, 
C=arc sin(0.884)--------> 180-62.1--------> C=117.9°

A+B+C=180°
som∠A = 180 - 45 - 117.9 = 17.1°
a=(sin A/sin B)*b--------> (sin 17.1/sin 45)*4------> 1.66

The second triangle has
m∠A = 17.1°,  m∠B = 45°,  m∠C = 117.9°,  a = 1.66,  b = 4,  c = 5

The answer is
 the number of possible triangles are 2