Let's analyze the problem from the image you provided.

The given problem asks for solving for $x$ and finding the angle measures for verification. The equations for the angles provided are as follows:

1. $m \angle AOB = (5x - 9)^\circ$
2. $m \angle BOC = (7x + 16)^\circ$
3. $m \angle COD = (2x + 12)^\circ$

From the diagram, we can see that points $A$, $B$, $C$, and $D$ seem to form angles around a point $O$. This suggests that the sum of these angles should equal 360 degrees since they are around a common vertex.

Therefore, we can write:

$m \angle AOB + m \angle BOC + m \angle COD = 360^\circ$

Substituting the expressions for the angles:

$(5x - 9) + (7x + 16) + (2x + 12) = 360$

Now, let's solve this equation.

Step-by-step:

$5x - 9 + 7x + 16 + 2x + 12 = 360$

Combine like terms:

$(5x + 7x + 2x) + (-9 + 16 + 12) = 360$

$14x + 19 = 360$

Subtract 19 from both sides:

$14x = 341$

Now, divide by 14:

$x = \frac{341}{14} \approx 24.36$

It seems there's a discrepancy with the given $x = 7$ in your screenshot, which might indicate a special case or rounding. Let's substitute $x = 7$ and calculate the angle measures to see if the results match.

Do you want me to proceed with substituting $x = 7$ into the angle equations to verify the results?