While studying analytic functions in complex analysis I found that if a function $f : \Bbb C \longrightarrow \Bbb C$ is an analytic function then if $\Bbb C$ is viewed as $\Bbb R^2$ then $f : \Bbb R^2 \longrightarrow \Bbb R^2$ is a differentiable vector valued function and in addition to that if $f(x,y) = \left (u(x,y), v(x,y) \right ),$ $x,y \in \Bbb R$ then $$Df(x,y) = \begin{pmatrix} \frac {\partial u} {\partial x} (x,y) & \frac {\partial u} {\partial y} (x,y)\\ -\frac {\partial u} {\partial y}(x,y) & \frac {\partial u} {\partial x}(x,y) \end{pmatrix}.$$
Now suppose $a+ib \in \Bbb C$ such that $f'(a+ib) \neq 0,$ where $f'(a+ib)$ denotes the complex derivative of $f$ at $a+ib.$ Then we have $$Df(a,b) \left [ \begin{pmatrix} \alpha_1 \\ \alpha_2 \end{pmatrix} \right ] = \left (\alpha_1 \frac {\partial u} {\partial x} (a,b) + \alpha_2 \frac {\partial u} {\partial y} (a.b) , -\alpha_1 \frac {\partial u} {\partial y} (a,b) + \alpha_2 \frac {\partial u} {\partial x} (a,b) \right ),\ \ \text {for any}\ \begin{pmatrix} \alpha_1 \\ \alpha_2 \end{pmatrix} \in \Bbb R^2.$$ If we see $Df(a,b) \left [ \begin{pmatrix} \alpha_1 \\ \alpha_2 \end{pmatrix} \right ]$ as a complex number then we have $$Df(a,b) \left [ \begin{pmatrix} \alpha_1 \\ \alpha_2 \end{pmatrix} \right ] = f'(a+ib) \cdot \alpha,\ \ \text {where}\ \alpha = \begin{pmatrix} \alpha_1 \\ \alpha_2 \end{pmatrix} \in \Bbb R^2.$$ So if $f'(a+ib) \neq 0$ then $Df(a,b)$ sends any complex number (or a vector in $\Bbb R^2$) to a complex number (or a vector in $\Bbb R^2$) whose modulus (or the length) is scaled by a factor $\left |f'(a+ib) \right |$ and rotated the vector through an additional angle $\arg \left (f'(a+ib)\right ).$
This is the geometric interpretation of the complex diffrentiation if the derivative doesn't vanish at a point.
In my lecture note a liitle remark has been made after giving this geometric aspect of an analytic function with non-vanishing derivative at a point. It states that
"If $\gamma_1$ and $\gamma_2$ be any two curves in the complex plane passing through the point $a+ib$ where $f'(a+ib) \neq 0$ then the angle between the curves at $a+ib$ is preserved by $f.$"
I don't get that point. How does the above implication follow from the geometric interpretation I mentioned above? Any help in this regard will be highly appreciated.
Thank you very much for reading.
EDIT $:$ What I see is that the linear operator $Df(a,b) : \Bbb R^2 \longrightarrow \Bbb R^2$ will preserve angle between any two curves in $\Bbb R^2.$ Is it helpful anyway in proving the required result?