Let [tex]\vec a,\vec b,\vec c[/tex] be vectors that point to [tex]A,B,C[/tex], respectively.
[tex]\vec a = 7\,\vec\imath+8\,\vec\jmath+2\,\vec k[/tex]
[tex]\vec b = 2\,\vec\imath + \vec\jmath + 2\,\vec k[/tex]
[tex]\vec c = -5\,\vec\imath-6\,\vec\jmath-6\,\vec k[/tex]
Then consider the directed line segments [tex]AB[/tex], [tex]BC[/tex], and [tex]CA[/tex], to which we'll assign the vectors
[tex]AB ~:~ \vec b - \vec a = -5\,\vec\imath - 7\,\vec\jmath[/tex]
[tex]BC ~:~ \vec c - \vec b = -7\,\vec\imath-7\,\vec\jmath-8\,\vec k[/tex]
[tex]CA ~:~ \vec a - \vec c = 12\,\vec\imath+14\,\vec\jmath+8\,\vec k[/tex]
whose lengths are
[tex]\|\vec b - \vec a\| = \sqrt{(-5)^2+(-7)^2} = \sqrt{74}[/tex]
[tex]\|\vec c - \vec b\| = \sqrt{(-7)^2+(-7)^2+(-8)^2} = \sqrt{162} = 9\sqrt2[/tex]
[tex]\|\vec a - \vec c\| = \sqrt{12^2+14^2+8^2} = \sqrt{404} = 2\sqrt{101}[/tex]
The angle at vertex [tex]A[/tex] is made by the directed segments [tex]AB[/tex] and [tex]AC[/tex], corresponding to [tex]\vec b-\vec a[/tex] and [tex]\vec c-\vec a[/tex]. Use the dot product identity to find the measure of [tex]\angle A[/tex].
[tex](\vec b - \vec a) \cdot (\vec c - \vec a) = \|\vec b - \vec a\| \|\vec c - \vec a\| \cos\left(m\angle A\right) \\\\ 158 = 2\sqrt{7474} \cos\left(m\angle A\right) \\\\ \cos\left(m\angle A\right) = \dfrac{79}{\sqrt{7474}} \\\\ m\angle A = \cos^{-1}\left(\dfrac{79}{\sqrt{7474}}\right) \approx \boxed{23.964^\circ}[/tex]
Similarly, the angle at [tex]B[/tex] is made by [tex]\vec a-\vec b[/tex] and [tex]\vec c-\vec b[/tex][/tex]. Then
[tex](\vec a-\vec b) \cdot (\vec c - \vec b) = \|\vec a - \vec b\| \|\vec c - \vec b\| \cos\left(m\angle B\right) \\\\ \cos\left(m\angle B\right) = -\dfrac{14}{3\sqrt{37}} \\\\ m\angle B = \cos^{-1}\left(-\dfrac{14}{3\sqrt{37}}\right) \approx \boxed{140.103^\circ}[/tex]
Do the same for [tex]C[/tex], or simply use the fact that the interior angles in any triangle sum to 180°. You should find
[tex]m\angle C = \cos^{-1}\left(\dfrac{41}{3\sqrt{202}}\right) = 180^\circ - m\angle A - m\angle B \approx \boxed{15.933^\circ}[/tex]
