If angles of arbitrary triangle add to 180$^\circ$, is there an isometry into the plane?

Question

Let $M$ be a 2-manifold with the property that for each $x\in M$ there exists a neighborhood $U_x$ around $x$ with the property that for any three points $A, B, C\in U_x$, the interior angle measure of $\triangle ABC$ is $180^\circ$. Does this necessarily imply that there exists an isometry $\phi: U_x\rightarrow \mathbb{R}^2$? If so, how could one go about proving its existence?

Answer 1

By the Gauss-Bonnet Theorem, applied to small geodesic triangles, the Gauss' curvature of $M$ is identically zero. This implies that $M$ is locally isometric to the euclidean plane.