  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$5\arcmin~(450 kpc) and
$\sim$12\arcmin~(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_2\sim1.44\pm0.16$ 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).
