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In this paper, we address the problem of determining the 2-D relative pose of pairs of communicating robots from (1) robot-to-robot distance measurements and (2) displacement estimates expressed in each robot's reference frame. Specifically, we prove that for nonsingular configurations, the minimum number of distance measurements required for determining all six possible solutions for the 3 degree-of-freedom (3-DOF) robot-to-robot transformation is 3. Additionally, we show that given four distance measurements, the maximum number of solutions is 4, while five distance measurements are sufficient for uniquely determining the robot-to-robot transformation. Furthermore, we present an efficient algorithm for computing the unique solution in closed form and describe an iterative least-squares process for improving its accuracy. Finally, we derive necessary and sufficient observability conditions based on Lie derivatives and evaluate the performance of the proposed estimation algorithms both in simulation and via experiments.