Interior angle sums of geodesic triangles in $\mathbf{S^2}\times\mathbf{R}$ and $\mathbf{H^2}\times\mathbf{R}$ geometries

In the present paper we study $\mathbf{S^2}\times\mathbf{R}$ and $\mathbf{H^2}\times\mathbf{R}$ geometries, which are homogeneous Thurston $3$-geometries. We analyse the interior angle sums of geodesic triangles in both geometries and we prove that in $\mathbf{S^2}\times\mathbf{R}$ space it can be larger than or equal to $\pi$ and in $\mathbf{H^2}\times\mathbf{R}$ space the angle sums can be less than or equal to $\pi$. This proof is a new direct approach to the issue and it is based on the projective model of $\mathbf{S^2}\times\mathbf{R}$ and $\mathbf{H^2}\times\mathbf{R}$ geometries described by E. Molnár in [7].