I renamed angles D, E, F as A, B and C
Also I have labeled the sides as a b c.
Look at the graphic I have attached.
cos (A) = (b^2 + c^2 -a^2) / 2bc
cos (A) = (6*6 + 8*8 -12*12) / (2 * 6 * 8)
cos (A) = (36 + 64 -144) / (96
(cos (A) = -44 / 96cos (A) = -0.4583333333
Then we look up the arc cosine of -0.4583333333 which equals 117.28 Degrees
We now have Angle A
From here, we can use the Law of Sines:
side a / sine (A) = side b / sine (B)
12 / sine (117.28) = 6 / sine (B)
12 / 0.88878 = 6 / sine (B)
sine (B) = 6 * .88878 / 12
sine (B) = 0.44439
We look up the arc sine of .44439 and it is
26.384 Degrees
Angle B = 26.384 Degrees
And Angle C is easily solved.
ALL triangles have angles that sum to 180 degrees.
180 = 117.28 + 26.384 + Angle C
Angle C = 180 -117.28 - 26.384
Angle C = 36.336 Degrees