Two lines intersect and form angles 4m-15 and m+75. Find m.

Since vertically opposite angles are equal, we can set the expressions for the two angles equal to each other.

$$4m - 15 = m + 75$$

To solve for $m$, we need to get all terms with $m$ on one side of the equation and all constant terms on the other side. Subtract $m$ from both sides and add 15 to both sides.

$$4m - m = 75 + 15$$

Combine the terms on each side.

$$3m = 90$$

Divide both sides by 3 to isolate $m$.

$$\frac{3m}{3} = \frac{90}{3}$$ $$m = 30$$

The value of $m$ is 30.

To check your answer, substitute $m=30$ back into the original angle expressions:
Angle 1: $4(30) - 15 = 120 - 15 = 105^\circ$
Angle 2: $30 + 75 = 105^\circ$
Since both angles are $105^\circ$, our value for $m$ is correct.