Given: [tex]\(\angle ABE\)[/tex] and [tex]\(\angle DBC\)[/tex] are vertical angles.

Prove: [tex]\(x=4\)[/tex]

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]. Therefore, we have:

[tex]\[ m \angle ABE = m \angle DBC \][/tex]

Substituting the given expressions, we get:

[tex]\[ 2x + 6 = x + 10 \][/tex]

Applying the subtraction property of equality:

[tex]\[ 2x + 6 - x = x + 10 - x \][/tex]

Simplifies to:

[tex]\[ x + 6 = 10 \][/tex]

Subtracting 6 from both sides, we get:

[tex]\[ x = 4 \][/tex]

Thus, we have proved that [tex]\(x=4\)[/tex].