https://file.aiquickdraw.com/mathbot/file/1727976483090nt884xbx.jpg

The problem presented in the image asks to find the values of \( x \) and \( y \) for the angles in the given diagram. Here is the breakdown:

The diagram consists of two intersecting lines forming vertical angles. We are given the following expressions for the angles:

1. Angle \( \angle EBC \) is labeled as \( 3x + 6^\circ \).
2. Angle \( \angle BDF \) is labeled as \( y^\circ \).
3. Angle \( \angle CAF \) is labeled as \( (x - 10)^\circ \).
4. Angle \( \angle DCF \) is labeled as \( (x - 11)^\circ \).

### Step 1: Use Vertical Angles
We know from geometry that vertical angles are congruent, meaning their measures are equal. Using this:

- \( \angle EBC = \angle DCF \), so:
  \[
  3x + 6 = x - 11
  \]
- \( \angle BDF = \angle CAF \), so:
  \[
  y = x - 10
  \]

### Step 2: Solve for \( x \)
First, solve the equation \( 3x + 6 = x - 11 \):
\[
3x - x = -11 - 6
\]
\[
2x = -17
\]
\[
x = -\frac{17}{2} = -8.5
\]

### Step 3: Solve for \( y \)
Now, use the equation \( y = x - 10 \) to find \( y \):
\[
y = -8.5 - 10 = -18.5
\]

### Final Answer:
- \( x = -8.5 \)
- \( y = -18.5 \)

Would you like more details or clarifications? Here are five questions to explore further:

1. Why are vertical angles always congruent?
2. What happens if the angles were supplementary instead of vertical?
3. How can you verify if your solutions for \( x \) and \( y \) are correct?
4. What is the importance of solving systems of equations in geometry?
5. How would the problem change if the angle expressions were different?

**Tip:** Always check your solutions by substituting the values back into the original equations.

Find the values of x and y given the angles: \( \angle EBC = 3x + 6 \), \( \angle BDF = y \), \( \angle CAF = (x - 10) \), \( \angle DCF = (x - 11) \).

Learn how to solve for x and y by applying the Vertical Angles Theorem and solving linear equations. This geometry problem involves intersecting lines and is suitable for students in Grades 8-10.

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