What 'preserves angles' actually means
You ended the last rung with a vivid one-line summary of the complex derivative: near a point z_0, a holomorphic function acts as one amplitwist — it amplifies every tiny arrow by the factor |f'(z_0)| and twists each one by the angle arg f'(z_0). This guide takes that infinitesimal picture and reads off its single most useful consequence: holomorphic maps preserve angles. But before we can prove that, we have to say carefully what the angle between two curves even is, because curves are not straight.
Picture two smooth curves through the same point z_0 — say a river and a road crossing at a bridge. At the crossing each curve has a tangent direction, the way it is heading at that instant, and the angle between the curves is defined to be the angle between those two tangent lines. So 'the angle two curves make' is really shorthand for 'the angle their tangent arrows make at the meeting point'. A map is said to preserve angles at z_0 if, whenever two curves cross there at some angle alpha, their images cross at the image point f(z_0) at the very same angle alpha.
There is a second, easily-missed half to the definition: a map is conformal if it preserves not only the size of the angle but also its sense — the direction of turning from the first arrow to the second, clockwise versus counterclockwise. A mirror reflection keeps the size of every angle but flips its sense; that is angle-preserving but anti-conformal, not conformal. Conformal means angles AND orientation are kept, and as you will see, holomorphy hands you both at once.
Why holomorphy keeps the angle: the two-arm argument
Here is the proof, and it is short because the amplitwist did all the heavy lifting. Let two smooth curves pass through z_0 with tangent directions theta_1 and theta_2 — these are the angles their tangent arrows make with the positive real axis. The angle between the curves is the difference theta_2 - theta_1. Now apply the map. Near z_0 the holomorphic f stretches every tiny arrow by |f'(z_0)| and rotates every one through the same fixed angle phi = arg f'(z_0). The stretching does nothing to directions. The rotation moves each tangent direction by the same phi.
before: tangent 1 at angle theta_1
tangent 2 at angle theta_2
angle between curves = theta_2 - theta_1
f rotates EVERY arrow by the same phi = arg f'(z_0):
after: image tangent 1 at angle theta_1 + phi
image tangent 2 at angle theta_2 + phi
angle between images = (theta_2 + phi) - (theta_1 + phi)
= theta_2 - theta_1 <-- unchangedThat is the whole argument. Because the common rotation phi adds to theta_1 and to theta_2 equally, it cancels in the difference theta_2 - theta_1, so the angle between the images equals the angle between the originals. And because phi is a genuine rotation (not a reflection), the turning sense from arm one to arm two is preserved too — so the map is conformal, not merely angle-preserving in size. This clean cancellation is the entire content of angle preservation; everything else in this rung is built on it.
Where the magic fails: critical points
The two-arm argument quietly assumed something we must now drag into the light: that f'(z_0) is not zero. The rotation angle phi = arg f'(z_0) only makes sense if f'(z_0) actually has an argument, and the complex number 0 has no argument at all — there is no direction in which 0 points. A point where f'(z_0) = 0 is called a critical point, and at a critical point the angle-preserving guarantee genuinely breaks. This is not a technicality to wave away; it is the next guide's whole subject.
The cleanest example is the squaring map f(z) = z^2 at the origin. Its derivative f'(z) = 2z vanishes at z = 0, so z = 0 is a critical point. Watch what happens to angles there. A ray leaving the origin at angle theta is the set of points r e^(i theta) for r > 0; squaring sends r e^(i theta) to r^2 e^(i 2 theta), a ray at angle 2 theta. Every angle measured at the origin is DOUBLED. Two rays that left 30 degrees apart arrive 60 degrees apart. The map is wildly un-conformal at exactly the one point where its derivative died.
This is the rule, not a fluke: if f has a zero of order m in its derivative at z_0 — meaning f behaves like a constant plus c (z - z_0)^(m+1) near there — then angles at z_0 are multiplied by m + 1. For z^2 the derivative has a simple zero, m = 1, and angles double, just as we saw. So the honest statement of the headline is: a holomorphic map is conformal at every point where f'(z_0) is non-zero, and it is precisely the isolated critical points, where f'(z_0) = 0, that you must flag and treat separately.
Why anyone cares: transplanting problems
Angle preservation would be a pretty curiosity if it were not also a powerful tool. The payoff is transplanting a problem: take a hard region — an oddly shaped domain in the plane — and find a conformal map carrying it to an easy region, usually a disk or a half-plane. Many physical laws are stated in terms of harmonic functions (think steady heat, ideal fluid flow, electrostatic potential), and harmonic-ness is governed by Laplace's equation, which a conformal change of variables leaves intact. So you solve the easy problem on the disk, then map the answer back, and the angles — the way streamlines meet boundaries, the way field lines cross equipotentials — come back correct because the map preserved them.
The explicit toolbox this rung will build
Knowing that conformal maps exist and preserve angles is one thing; having a stock of concrete ones to reach for is another. The star of this rung is the Mobius transformation, a map of the form (a z + b) / (c z + d) with a d - b c not zero. These are the simplest non-trivial holomorphic maps, they are conformal wherever they are defined, and they have a remarkable closure property you will prove soon: they send every line-or-circle to another line-or-circle. They are flexible enough that you can specify where any three points should go and a unique Mobius map obliges — the three-point principle two guides ahead.
Two named maps will recur because they solve recurring jobs. The Cayley transform w = (z - i) / (z + i) is the workhorse that carries the upper half-plane conformally onto the unit disk — exactly the transplant you want when a problem lives on a half-plane but the disk is easier. The Joukowski map w = (z + 1/z) / 2 famously bends a circle into an airfoil shape and underlies the classical theory of lift on a wing — a vivid reminder that this geometry pays rent in aerodynamics, not just in proofs.
Stepping further back, there is a breathtaking theorem promising that the search for a transplant almost never fails. The Riemann mapping theorem says that ANY simply connected region that is not the entire plane can be mapped conformally onto the unit disk. Be honest about two caveats, though: it excludes the whole plane (there is no conformal map from the plane onto the disk, a fact Liouville's theorem from the last rung forces), and it is non-constructive — it guarantees a map exists but does not hand you a formula. That gap between 'exists' and 'here it is' is exactly why the explicit toolbox above matters so much.