Determine whether the triangle has no/one/two solutions. Then Find and round measures of sides to the nearest tenth and measures of the angles to the nearest degree.

A= 50 degrees, a=34, b=40

sinA/a = sinB/b

sinB/40 = sin50°/34
sinB = 40sin50/34 = .901229
B = 64.32 or B = 115.68

Case1:
A = 50, B=64.32 , C = 65.68
c/sinC = a/sinA
c= 34sin65.68 /sin50 =40.445

case2:
A=50 , B = 115.68 C = 14.32°

c = 34sin14.32/sin50 = 10.98 = appr 11.0

Case1
A = 50° , B= 64° , C = 66°
a=34 , b=40 , c=40.4

Case2
A=50° , B= 116°, C= 14°
a=34, b=40 , c = 11.0

To determine the number of solutions for a triangle, we can apply the Triangle Inequality Theorem. According to this theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

Let's check if the given triangle satisfies the Triangle Inequality Theorem:
a + b > c
34 + 40 > c
74 > c

b + c > a
40 + c > 34
c > -6

a + c > b
34 + c > 40
c > 6 - 34
c > -28

The measures of the three sides are:
a = 34
b = 40
c > -28

Since the length of a side cannot be negative, we conclude that this triangle has two solutions. Now, let's move on to finding the measures of the sides and angles.

To find the measure of side c, we can use the Law of Cosines. This law states that in any triangle, the square of one side is equal to the sum of the squares of the other two sides, minus twice the product of the two sides, multiplied by the cosine of the included angle.

c^2 = a^2 + b^2 - 2ab * cos(A)

Substituting the given values:
c^2 = 34^2 + 40^2 - 2*34*40 * cos(50)

Now we can calculate c:
c^2 = 1156 + 1600 - 2720 * cos(50)
c^2 = 2756 - 2720 * cos(50)
c^2 ≈ 439.534

Taking the square root of both sides:
c ≈ √439.534
c ≈ 20.95

To find angles B and C, we can use the Law of Sines. This law states that the ratio of the length of a side to the sine of the angle opposite that side is constant for all sides and angles in a given triangle.

sin(B)/b = sin(A)/a

Substituting the given values:
sin(B)/40 = sin(50)/34

Now we can find sin(B):
sin(B) = (40 * sin(50))/34
sin(B) ≈ 0.785

To find angle B, we can take the inverse sine of sin(B):
B ≈ arcsin(0.785)
B ≈ 51.2 degrees (rounded to the nearest degree)

To find angle C, we can subtract angles A and B from 180 degrees since the sum of angles in a triangle is always 180 degrees:
C ≈ 180 - A - B
C ≈ 180 - 50 - 51.2
C ≈ 78.8 degrees (rounded to the nearest degree)

To summarize, the measures of the sides rounded to the nearest tenth are:
a ≈ 34
b ≈ 40
c ≈ 20.9

The measures of the angles rounded to the nearest degree are:
A ≈ 50 degrees
B ≈ 51 degrees
C ≈ 79 degrees