See the image attachment for how I labeled the points and segments.
Based on the image attachment, we have a = 3.51, b = 5.96, c = 4.2
I'll use the law of cosines for each sub-problem
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Solving for angle A
a^2 = b^2 + c^2 - 2*b*c*cos(A)
(3.51)^2 = (5.96)^2 + (4.2)^2 - 2*(5.96)*(4.2)*cos(A)
12.3201 = 17.64 + 35.5216 - 50.064*cos(A)
12.3201 = 53.1616 - 50.064*cos(A)
12.3201 - 53.1616 = 53.1616 - 50.064*cos(A)-53.1616
-40.8415 = -50.064*cos(A)
(-40.8415)/(-50.064) = (-50.064*cos(A))/(-50.064)
0.815785794183445 = cos(A)
cos(A) = 0.815785794183445
arccos(cos(A)) = arccos(0.815785794183445)
A = 35.334866310355
Angle A is roughly 35.33 degrees
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Solving for angle B
b^2 = a^2 + c^2 - 2*a*c*cos(B)
(5.96)^2 = (3.51)^2 + (4.2)^2 - 2*(3.51)*(4.2)*cos(B)
35.5216 = 12.3201 + 17.64 - 29.484*cos(B)
35.5216 = 29.9601 - 29.484*cos(B)
35.5216 - 29.9601 = 29.9601 - 29.484*cos(B)-29.9601
5.5615 = -29.484*cos(B)
(5.5615)/(-29.484) = (-29.484*cos(B))/(-29.484)
-0.188627730294397 = cos(B)
cos(B) = -0.188627730294397
arccos(cos(B)) = arccos(-0.188627730294397)
B = 100.872710934509
Angle B is roughly 100.87 degrees
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Solving for angle C
c^2 = a^2 + b^2 - 2*a*b*cos(C)
(4.2)^2 = (3.51)^2 + (5.96)^2 - 2*(3.51)*(5.96)*cos(C)
17.64 = 12.3201 + 35.5216 - 41.8392*cos(C)
17.64 = 47.8417 - 41.8392*cos(C)
17.64 - 47.8417 = 47.8417 - 41.8392*cos(C)-47.8417
-30.2017 = -41.8392*cos(C)
(-30.2017)/(-41.8392) = (-41.8392*cos(C))/(-41.8392)
0.721851756247729 = cos(C)
cos(C) = 0.721851756247729
arccos(cos(C)) = arccos(0.721851756247729)
C = 43.7924227551359
Angle C is roughly 43.79 degrees
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Check out the attachment to see the final answers in visual form.
