This question is from physics, but I think the answer is more-so fundamentally a fact of mathematics, rather than physics which is why I'm posting it here.
My textbook, Solid-State Physics, Fluidics, and Analytical Techniques in Micro- and Nanotechnology, by Madou, presents the following image and explanation in a section on x-ray diffraction and Laue equations:
Bragg’s law is equivalent to the Laue equations in one dimension as can be appreciated from an inspection of Figures 2.24 and 2.25, where we use a two-dimensional crystal for simplicity. Suppose that vector $\Delta \mathbf{k}$ in Figure 2.24 satisfies the Laue condition; because incident and scattered waves have the same magnitude (elastic scattering), it follows that incoming ($\mathbf{k}_0$) and reflected rays ($\mathbf{k}$) make the same angle $\theta$ with the plane perpendicular to $\Delta \mathbf{k}$.
So this passage seems to be saying that, if two vectors $\mathbf{k}$ and $\mathbf{k}_0$ satisfy the condition $\Delta \mathbf{k} = \mathbf{k} - \mathbf{k}_0$, where $\mathbf{k}_0$ is an incoming ray, and $\mathbf{k}$ is a reflected, outgoing ray, and if these rays have the same magnitude, then it must be that the rays make the same angle $\theta$ with the plane perpendicular to $\Delta \mathbf{k}$. Is this a mathematical fact? And if so, then does anyone have a proof of this?
I would greatly appreciate it if people would please take the time to clarify this.
Your statement is not generally true. But in the text there is one more condition that you did not use: "satisfies the Laue condition". It means that $\Delta\textbf{k}$ is perpendicular to that plane. It's easy to show that in an isosceles triangle, the perpendicular from the vertex where the two equal sides meet onto the other side is also the bisector of the angle between the two equal sides.