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.).