This paper studies the number \(p_{3}(B)\) of triangles in a non-trivial arrangement \(B\) of \(n\) pseudolines in the Euclidean plane. The main results are: (1) If \(B\) is simple, then \(p_{3}(B)\geq n-2\) with equality possible for all \(n\geq 3\). (2) If \(n\geq 6\), then \(p_{3}(B)\geq 2n/3\) with equality possible for all \(n\equiv 0 \text{mod} 3\). For straight line arrangements, the inequality of (1) is known to hold also in case \(B\) is not simple. Thus the following corollary is obtained: If \(p_{3}(B)