OBM 1986 Problem 3

The Poincare plane is a half-plane bounded by a line $R$ . The lines are taken to be (1) the half-lines perpendicular to $R$ , and (2) the semicircles with center on $R$ . Show that given any line $L$ and any point $P$ not on $L$ , there are infinitely many lines through $P$ which do not intersect $L$ . Show that if $ABC$ is a triangle, then the sum of its angles lies in the interval $(0, \pi)$ .