It is given that [tex]$\angle ABE$[/tex] and [tex]$\angle DBC$[/tex] are vertical angles. By the vertical angles theorem, [tex]$\angle ABE$[/tex] is congruent to [tex]$\angle DBC$[/tex]. By the definition of congruence, the measure of [tex]$\angle ABE$[/tex] must equal the measure of [tex]$\angle DBC$[/tex]. If we let the measure of [tex]$\angle ABE$[/tex] be [tex]$2x + 6$[/tex] and the measure of [tex]$\angle DBC$[/tex] be [tex]$x + 10$[/tex], then by the substitution property of equality, we have [tex]$2x + 6 = x + 10$[/tex]. Applying the subtraction property of equality, we subtract [tex]$x$[/tex] from both sides to get [tex]$x + 6 = 10$[/tex]. Finally, subtracting [tex]$6$[/tex] from both sides, we obtain that [tex]$x = 4$[/tex].