Edit: It seems you want to find the measure of the angle at vertex C.
Consider two vectors x and y, the first of which begins at vertex C and ends at B, and the other begins at C and ends at A. We want to find the angle between x and y. If γ (gamma) is the measure of this angle, then
x • y = ||x|| ||y|| cos(γ)
Translating these vectors so that they start at the origin O, we get
C to B : x = (-2, 2) - (6, -2) = (-8, 4)
C to A : y = (6, 4) - (6, -2) = (0, 6)
Compute the dot product of x and y :
x • y = (-8, 4) • (0, 6) = 24
Compute their magnitudes:
||x|| = √((-8)² + 4²) = 4√5
||y|| = √(0² + 6²) = 6
Solve for γ :
cos(γ) = (x • y) / (||x|| ||y||)
cos(γ) = 24/(24√5)
cos(γ) = 1/√5
γ = arccos(1/√5) ≈ 63.4349°