Step-by-step explanation:
I don't see a triangle here.
but based on the given information we can rely on the law of cosine (which I call the general Pythagoras) :
c² = a² + b² - 2ab×cos(C)
where C is the angle opposite to the side c, a and b are the other 2 sides.
and we can rely on the law of sine :
a/sinA = b/sinB = c/sinC
where A, B, C are the angles opposite of the corresponding sides a, b, c.
also remember, the sum of all angles in a triangle is always 180°.
so, we can start with
c/sinC = b/sinB
27/sin(31) = 38/sin(B)
sin(B) = 38×sin(31)/27 = 0.724868402...
B = 46.45790151...° ≈ 46.5°
or (as I cannot see the model triangle) it can be an obese angle 180 - 46.45790151... = 133.5420985... ≈ 133.5°
this is because sine is positive in the first and second quadrant, and the sine of the angles x and 180-x have the same value.
that gives us for the angle A :
180 = 31 + 46.5 + A
A = 102.5°
or
180 = 31 + 133.5 + A
A = 15.5°
and for side a
a² = b² + c² - 2bc×cos(A) =
= 38² + 27² - 2×38×27×cos(102.5) =
= 1444 + 729 - 2052×-0.216439614... =
= 2,617.134088...
a = 51.15793279... ≈ 51.2
or
a² = b² + c² - 2bc×cos(A) =
= 38² + 27² - 2×38×27×cos(15.5) =
= 1444 + 729 - 2052×0.963630453... =
= 195.63031...
a = 13.98679055... ≈ 14.0
so, the complete triangle is
A = 102.5°
or
A = 15.5°
B = 46.5°
or
B = 133.5°
C = 31°
a = 51.2
or
a = 14.0
b = 38
c = 27
