Given that Ray BE bisects Angle ABD:

 

If the measure of Angle ABE = (6x + 2) degrees and the measure of Angle DBE = (8x - 14) degrees.

 

Find the measure of Angle ABE:               degrees

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.

Given that Ray BE bisects Angle ABD: If the measure of Angle ABE = (6x + 2) degrees and the measure of Angle DBE = (8x - 14) degrees, find the measure of Angle ABE.

Learn how to find the measure of Angle ABE when Ray BE bisects Angle ABD. This step-by-step solution covers the concepts of angle bisectors and solving linear equations. Suitable for students in Grades 7-10.

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