 Solution

For a regular smooth curve $z=z_j(t)$ passing through $z_0$, the complex tangent of its image is $d{w}_j/dt=f^{\prime }(z_0)z_j^{\prime }(t)$. When $f^{\prime }(z_0)\ne 0$, both tangents are multiplied by the same nonzero complex number. Multiplication rescales lengths by $|f^{\prime }(z_0)|$ and rotates every direction by $\arg f^{\prime }(z_0)$, so the difference of tangent arguments is unchanged. Hence the analytic function defines a conformal map at $z_0$: oriented angles, and therefore ordinary angles between the curves, are preserved. The nonzero derivative condition also ensures that the image tangents are nonzero.