The two angles marked in the diagram are vertical. Since vertical angles are congruent, we have:

\begin{align*} 5x-2^\circ & = 123^\circ \\[3pt] 5x & = 125^\circ \\[3pt] x & = 25^\circ \end{align*}

The two angles marked in the diagram are vertical. Since vertical angles are congruent, we have:

\begin{align} 3x+3^\circ & = 60^\circ \\[3pt] 3x & = 57^\circ \\[3pt] x & = 19^\circ \end{align}

The two angles marked in the diagram are vertical. Since vertical angles are congruent, we have:

\begin{align} 8x-25^\circ& = 4x-3^\circ \\[5pt] 8x& = 4x +22^\circ \\[5pt] 4x& = 22^\circ \\[5pt] x & = 5.5^\circ \end{align}

Now that we know $x=5.5^\circ,$ we can substitute into one of the expressions to calculate the measure of the angles. Let's substitute into the expression $4x-3^\circ\mathbin{:}$

\[ 4(5.5^\circ) - 3^\circ = 19^\circ. \]

Therefore, both angles have a measure of $19^\circ.$

The two angles marked in the diagram are vertical. Since vertical angles are congruent, we have:

\begin{align} 5x+23^\circ& = 7x-1^\circ \\[5pt] 5x+24^\circ& = 7x \\[5pt] 24^\circ& = 2x \\[5pt] 12^\circ& = x \\[5pt] x & = 12^\circ \end{align}

Now that we know $x=12^\circ,$ we can substitute into one of the expressions to calculate the measure of the angles. Let's substitute into the expression $5x+23^\circ\mathbin{:}$

\[ 5(12^\circ) + 23^\circ = 83^\circ. \]

Therefore, both angles have a measure of $83^\circ.$

Since $\angle AOB$ is straight, $m\angle AOB = 180^\circ.$ So by the angle addition postulate, we have

\[ m \angle BOC + m \angle COA = 180^\circ. \]

Substituting the given expressions into the above, we can solve for $x$ as follows:

\begin{align*} (2x-15^\circ) + x & = 180^\circ \\[5pt] 3x-15^\circ& = 180^\circ \\[5pt] 3x & = 195^\circ \\[5pt] x & = 65^\circ \end{align*}

Now that we know $x=65^\circ,$ we can calculate $m \angle COA$ as follows: \[ m \angle COA = x = 65^\circ \]

Finally, since the angles $\angle BOD$ and $\angle COA$ are vertical angles, they have the same measure: \[ m \angle BOD = m \angle COA = 65^\circ \]

Since $\angle AOB$ is straight, $m\angle AOB = 180^\circ.$ So by the angle addition postulate, we have

\[ m \angle COB + m \angle COA = 180^\circ. \]

Substituting the given expressions into the above, we can solve for $x$ as follows:

\begin{align*} x + 3x & = 180^\circ \\[5pt] 4x & = 180^\circ \\[5pt] x & = 45^\circ \end{align*}

Now that we know $x=45^\circ,$ we can calculate $m \angle COA$ as follows: \[ m \angle COA = 3(45^\circ) = 135^\circ \]

Finally, since the angles $\angle BOD$ and $\angle COA$ are vertical angles, they have the same measure:

\[ m \angle BOD = m \angle COA = 135^\circ \]

We have two pairs of vertical angles, and we know that vertical angles are congruent and therefore have the same measure. So, we have

\begin{align*} m \angle 1 &= m \angle 2, \\[3pt] m \angle 3 &= m \angle 4. \end{align*}

We are told that $m \angle 1 = 60^\circ$ and $m \angle 4 = 80^\circ,$ so we have

\begin{align*} 60^\circ &= m \angle 2, \\[3pt] m \angle 3 &= 80^\circ. \end{align*}

Therefore, we obtain

\begin{align*} 4 m \angle 2 + 2 m \angle 3 & = 4(60^\circ) + 2(80^\circ) \\[3pt] & = 240^\circ + 160^\circ \\[3pt] & = 400^\circ. \end{align*}

We have two pairs of vertical angles, and we know that vertical angles are congruent and therefore have the same measure. So, we have

\begin{align*} m \angle 1 &= m \angle 2, \\[3pt] m \angle 3 &= m \angle 4. \end{align*}

We are told that $m \angle 1 = 115^\circ$ and $m \angle 4 = 55^\circ,$ so we have

\begin{align*} 115^\circ &= m \angle 2, \\[3pt] m \angle 3 &= 55^\circ. \end{align*}

Therefore, we obtain

\begin{align} m \angle 2 - m \angle 3 & = 115^\circ - 55^\circ \\[3pt] &= 60^\circ. \end{align}