If angle ABF =8w-6 and angle ABE =2w+22 and Angle ABE and Angle EBF are congruent and equal angle ABF what’s the measure of angle EBF

We are given the following information:

- \( \angle ABF = 8w - 6 \)
- \( \angle ABE = 2w + 22 \)
- \( \angle ABE \) and \( \angle EBF \) are congruent and both equal to \( \angle ABF \).

Since \( \angle ABE = \angle EBF = \angle ABF \), we can set these angles equal to each other.

### Step 1: Set up the equations

We have:
\[
\angle ABE = \angle ABF = 8w - 6
\]
Thus, we can set \( 2w + 22 = 8w - 6 \).

### Step 2: Solve for \( w \)

Now, solve for \( w \):
\[
2w + 22 = 8w - 6
\]
Subtract \( 2w \) from both sides:
\[
22 = 6w - 6
\]
Add 6 to both sides:
\[
28 = 6w
\]
Divide both sides by 6:
\[
w = \frac{28}{6} = \frac{14}{3}
\]

### Step 3: Calculate the measure of \( \angle ABF \), \( \angle ABE \), and \( \angle EBF \)

Since \( w = \frac{14}{3} \), substitute it back into \( \angle ABF = 8w - 6 \) to find the measure of these angles:
\[
\angle ABF = 8 \times \frac{14}{3} - 6 = \frac{112}{3} - 6 = \frac{112}{3} - \frac{18}{3} = \frac{94}{3} \approx 31.33^\circ
\]

Therefore, the measure of \( \angle EBF = 31.33^\circ \).

### Final Answer:
The measure of \( \angle EBF \) is approximately \( 31.33^\circ \).

Would you like further details or clarification on any step? Here are some related questions:

1. How do you set up angle relationships in congruent angles?
2. What is the importance of solving for \( w \) in such problems?
3. What are common properties of congruent angles?
4. How would you solve if angles were not congruent?
5. What does the angle sum property of triangles contribute to angle problems?

**Tip**: Always check if angles are congruent or supplementary in geometric problems; this often simplifies the equations.

If angle ABF = 8w - 6 and angle ABE = 2w + 22 and Angle ABE and Angle EBF are congruent and equal angle ABF, what’s the measure of angle EBF?

Find the measure of angle EBF when given that angles ABE and EBF are congruent and equal to angle ABF. This problem uses algebraic equations and geometric properties of congruent angles. Suitable for students in Grades 8-10.

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