= Solution
The <Cauchy-Riemann equations> give
$$
\nabla u\cdot\nabla v=u_xv_x+u_yv_y=u_x(-u_y)+u_yu_x=0.
$$
At a regular point, each level curve has tangent perpendicular to its nonzero <gradient>. The two <gradients>, and hence the two tangents, are orthogonal. Therefore \b[regular level curves meet at right angles].
The qualification is necessary: $|\nabla u|=|\nabla v|=|f'(z)|$, so at a critical point both <gradients> vanish and this argument gives no tangent direction. For example, for $f(z)=z^2$, the zero level curves of the real and imaginary parts are diagonals and coordinate axes; they meet at $45^\circ$ at zero. This is the regular-point statement of <orthogonal level curves of harmonic conjugates>.
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