The vertex where the two triangles touch forms a pair of vertical angles, so that the angles on opposites sides of the vertex are congruent. You're shown that two legs of both triangles are congruent. All this tells you that the two triangles are isosceles, and furthermore that they are similar because of the congruence of their "base" angles.
This means
[tex]82^\circ=m\angle 2[/tex]
[tex]\implies82=10x+12[/tex]
[tex]\implies10x=70[/tex]
[tex]\implies\boxed{x=7}[/tex]
