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The image shows an angle diagram with three labeled points: \( A \), \( B \), and \( C \), where two angles are given:

- Angle \( \angle ABD = (2x + 23)^\circ \)
- Angle \( \angle DBC = (9x - 5)^\circ \)

Since the two angles \( \angle ABD \) and \( \angle DBC \) form a straight line (which equals 180°), we can write the equation:

\[
(2x + 23) + (9x - 5) = 180
\]

Let's solve for \( x \).

### Step-by-step Solution:

1. Combine like terms:
\[
(2x + 9x) + (23 - 5) = 180
\]
\[
11x + 18 = 180
\]

2. Subtract 18 from both sides:
\[
11x = 162
\]

3. Solve for \( x \) by dividing both sides by 11:
\[
x = \frac{162}{11} \approx 14.73
\]

Thus, \( x \approx 14.73 \).

### Now, let's find the angles:

1. \( \angle ABD = 2x + 23 = 2(14.73) + 23 = 29.46 + 23 = 52.46^\circ \)
2. \( \angle DBC = 9x - 5 = 9(14.73) - 5 = 132.57 - 5 = 127.57^\circ \)

### Check:

\[
52.46^\circ + 127.57^\circ = 180^\circ
\]

So, the solution is consistent.

Would you like more details on the process? Here are some related questions:

1. What if the angles didn't form a straight line?
2. Can you explain angle relationships in a triangle?
3. How can I verify the consistency of solutions in geometry problems?
4. What are the applications of this angle relationship concept?
5. How does the straight angle property help solve equations?

**Tip:** Always double-check the sum of your angles in geometry problems to ensure they meet the required condition (e.g., straight line = 180°, triangle = 180°, etc.).

Find the value of x and the angles ABD and DBC if (2x + 23)° and (9x - 5)° form a straight line.

Learn how to solve a problem involving the angles ABD and DBC, given as (2x + 23)° and (9x - 5)°, which form a straight line. The step-by-step solution explains the use of algebra and geometry concepts, making it suitable for students in Grades 8-10.

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