I am given the following: Let $C$ be a simple closed curve in $\mathbb{R}^2$ and let $p \in C$. Show that there exists a line in $\mathbb{R}^2$, passing through $p$, that is not tangent to $C$ at any point.
First, we can assume that $p$ is the origin. My first attempt was to form the map $C \to \mathbb{R}P^1$ that maps $x \in C$ to the line joining $x$ and $p$. My hope was that the critical points of this map would correspond to tangent lines. Then I could use Sard's Theorem. I couldn't show that. My next thought was that we get a map $S^1 \to C \to \mathbb{R}P^1$ and maybe I could show something there. But, I could not think of anything. Any help is appreciated.
I guess we assume the curve having a non-singular $C^1$ parametrization, in which case your idea seems fine. Let $t\in S^1$ be the parameter and take e.g. $\theta(t)$ to be the angle (mod $2\pi$) of the line you mention wrt some fixed line. The critical values $\{\theta(t): \theta'(t)=0\}$ correspond to tangents and are of zero measure.