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The problem asks to find the measure of \(\angle EBF\) given:

- \( m\angle ABF = 8s - 6 \)
- \( m\angle ABE = 2(s + 1) \)

We know that \(\angle ABE + \angle EBF = \angle ABF\), so:

\[
m\angle ABE + m\angle EBF = m\angle ABF
\]

Substitute the given expressions into the equation:

\[
2(s + 1) + m\angle EBF = 8s - 6
\]

Now we solve for \( m\angle EBF \).

First, simplify \(2(s + 1)\):

\[
2(s + 1) = 2s + 2
\]

Now, substitute this back into the equation:

\[
2s + 2 + m\angle EBF = 8s - 6
\]

Solve for \( m\angle EBF \):

\[
m\angle EBF = (8s - 6) - (2s + 2)
\]
\[
m\angle EBF = 8s - 6 - 2s - 2
\]
\[
m\angle EBF = 6s - 8
\]

Thus, the measure of \(\angle EBF\) is \(6s - 8\).

Would you like further clarification or any additional details? Here are five related questions:

1. How do you solve for \(s\) if additional information about angles is given?
2. What happens if the angles form a straight line, and the sum equals 180°?
3. How would you solve this if the equation involved quadratic terms?
4. What methods can be used to check if the result is correct?
5. How would the solution change if \(m\angle ABF = m\angle ABE\)?

**Tip**: Always substitute values carefully to avoid calculation errors when simplifying algebraic expressions.

If m∠ABF = 8s - 6 and m∠ABE = 2(s + 1), find m∠EBF.

In this problem, we find the measure of ∠EBF given algebraic expressions for m∠ABF and m∠ABE. This solution involves using the angle addition formula and simplifying the resulting equation. Suitable for students in Grades 8-10.

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