Since Ray $BE$ bisects Angle $ABD$, it means that:

$\angle ABE = \angle DBE$

Given the expressions for the angles:

$\angle ABE = 6x + 2$ $\angle DBE = 8x - 14$

Since $\angle ABE = \angle DBE$, we can set the two expressions equal to each other:

$6x + 2 = 8x - 14$

Now, solve for $x$:

1. Subtract $6x$ from both sides: $2 = 2x - 14$

2. Add 14 to both sides: $16 = 2x$

3. Divide by 2: $x = 8$

Now, substitute $x = 8$ back into the expression for $\angle ABE$:

$\angle ABE = 6(8) + 2 = 48 + 2 = 50 \text{ degrees}$

Thus, the measure of $\angle ABE$ is 50 degrees.

Would you like further details or clarification on any step?

Here are 5 questions that could expand on this concept:

1. What is the measure of $\angle ABD$ given this information?
2. How do you verify the accuracy of the calculated angles?
3. How would the problem change if $BE$ did not bisect $\angle ABD$?
4. Can you generalize this method for other angle bisectors?
5. What are the key properties of an angle bisector?

Tip: Always check for relationships between angles, such as bisectors, when solving geometric problems.