We present the results of deep 140 ks Suzaku X-ray observations of the north-east (NE) radio relic of the merging galaxy cluster Abell2255. The temperature structure of Abell2255 is measured out to 0.9 times the virial radius (1.9 Mpc) in the NE direction for the first time. The Suzaku temperature map of the central region suggests a complex temperature distribution, which agrees with previous work. Additionally, on a larger-scale, we confirm that the temperature drops from 6 keV around the cluster center to 3 keV at the outskirts, with two discontinuities at {it r}$sim$5arcmin~(450 kpc) and $sim$12arcmin~(1100 kpc) from the cluster center. Their locations coincide with surface brightness discontinuities marginally detected in the XMM-Newton image, which indicates the presence of shock structures. From the temperature drop, we estimate the Mach numbers to be ${cal M}_{rm inner}sim$1.2 and, ${cal M}_{rm outer}sim$1.4. The first structure is most likely related to the large cluster core region ($sim$350--430 kpc), and its Mach number is consistent with the XMM-Newton observation (${cal M}sim$1.24: Sakelliou & Ponman 2006). Our detection of the second temperature jump, based on the Suzaku key project observation, shows the presence of a shock structure across the NE radio relic. This indicates a connection between the shock structure and the relativistic electrons that generate radio emission. Across the NE radio relic, however, we find a significantly lower temperature ratio ($T_1/T_2sim1.44pm0.16$ corresponds to~${cal M}_{rm X-ray}sim1.4$) than the value expected from radio wavelengths, based on the standard diffusive shock acceleration mechanism ($T_1/T_2>$ 3.2 or ${cal M}_{rm Radio}>$ 2.8).
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