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In a recent paper Freese and Murphy said a complete, convex, externally convex metric space has the vertical angle property provided for each four of its distinct points p, q, r, s, if m is a midpoint of p and q and of r and s, then pr = qs. In this paper we say a line L is parallel to a line N in such a space provided L and N contain points p, r, and q, s, respectively, such that the segments S(p, q) and S(r, s) have a common midpoint m. We further assume that if line L is parallel to line N and line N is parallel to line R, then L is parallel to R. The main result of this paper is that such a space is a real strictly convex Banach space. Since real strictly convex Banach spaces have all of the above properties, the characterization is then complete. The problem of characterizing Banach spaces among the class of complete, convex, externally convex metric spaces has been solved by various authors, see [1, 2, 4, 6 and 7]. (It is well known that a complete, convex, externally convex metric space contains a (metric) line for each pair of its distinct points, see [3].) The purpose of this paper is to show that the transitive property of parallelism does this. Of course, before we can talk about the transitive property of parallel lines, we need some way of defining what it means to say (metric) lines are parallel in a metric space. The vertical angle property of Freese and Murphy [5] provides a way. A metric space has the vertical angle property, VAP, provided for each four of its distinct points p, q, r, s, if m is a midpoint of p and q and of r and s, then pr = qs. Let M be a complete, convex, externally convex metric space with VAP. We will say a line L1 is parallel to a line L2 provided L1 and L2 contain points p, r and q, s, respectively, such that the segments S( p, q) and S(r, s) have a common midpoint m. We will say that a metric space M has the transitive property of parallel lines, TPPL, provided if L1 is parallel to L2 and L2 is parallel to L3, then L1 is parallel to L3. Throughout the remainder of this paper, M will denote a complete, convex, externally convex metric space that has VAP and TPPL. Our first theorem is proved in [5], but we include a proof here in order to make the paper complete. THEOREM 1. Each two distinct points of M lie on a unique metric line. Received by the editors September 5, 1984 and, in revised form, February 1, 1985. 1980 Mathematics Subject Classification. Primary 51K05, 51F99. Kev words and phrases. Banach space, convex, externally convex, metric space, strictly convex, transitive property of parallel lines, vertical angle property. ?1985 American Mathematical Society 0002-9939/85 $1.00 + $.25 per page

The transitive property of parallel lines is a characteristic property of real strictly convex Banach spaces