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In this diagram, we observe three angles formed by intersecting lines at points \( B \), \( C \), and \( E \). These angles are labeled in terms of \( x \):

1. \( \angle ABE = (17x + 2)^\circ \),
2. \( \angle ABC = (15x + 20)^\circ \),
3. \( \angle ECD = (9x + 10)^\circ \).

To solve for \( x \), we need to apply the property that the sum of the angles along a straight line is \( 180^\circ \). Specifically, we are interested in:

- \( \angle ABE \) and \( \angle ABC \) forming a straight line (linear pair),
- \( \angle ECD \) and its supplementary angle, which we might deduce based on other given information.

Let me solve this step by step.The value of \( x \) is \( \frac{79}{16} \) (approximately 4.94).

Now, using this value of \( x \), the angles are calculated as follows:

1. \( \angle ABE = \frac{1375}{16} \approx 85.94^\circ \),
2. \( \angle ABC = \frac{1505}{16} \approx 94.06^\circ \),
3. \( \angle ECD = \frac{871}{16} \approx 54.44^\circ \).

Would you like further details or clarification on any of the steps?

Here are some related questions for further exploration:

1. What other properties can we infer from the intersecting lines in the diagram?
2. How do supplementary angles help in solving such problems?
3. Could you verify the sum of angles in a triangle with these results?
4. What role do linear pairs play in geometry?
5. How does solving for variables in angles extend to more complex figures?

**Tip:** Always check for supplementary angles when dealing with intersecting lines to simplify calculations.

In the diagram, solve for x given the angles: \( \angle ABE = (17x + 2)^\circ \), \( \angle ABC = (15x + 20)^\circ \), and \( \angle ECD = (9x + 10)^\circ \).

This problem involves solving for x using properties of angles and linear pairs. The angles in the diagram are labeled as (17x + 2)°, (15x + 20)°, and (9x + 10)°. Learn how to use the sum of angles on a straight line and supplementary angle theorems to find the solution.

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