Mathematics Question and Solution
Calculating Alpha, let it be x.
a = 180-24-54
a = 180-78
a = 102°
Let the side length opposite/facing angle 24° (the shortest length) of the inscribed triangle with angles 24°, 54° and 102° be 1 unit.
It implies;
(1/sin24) = (b/sin102)
b = 2.40486717237 units.
(1/sin24) = (c/sin54)
c = 1.98904379074 units.
Therefore;
d² = 1.98904379074²+2.40486717237²-2*1.98904379074*2.40486717237cos(24+24)
d = 1.82709091528 units.
(1.82709091528/sin(24+24)) = (2.40486717237/sine)
e = asin(0.97814760073)
e = 78°
Angle x (Alpha) is;
x+e = a
x+78 = 102
x = 102-78
x = 24°
Or
78+54+x+x = 180
132+2x = 180
2x = 180-132
2x = 48
x = 24°
