Given that the lines $WX$ and $YZ$ intersect at point $V$ and are perpendicular to each other, the angles formed by the intersection are right angles, i.e., $\angle WVY$ and $\angle XVY$ are $90^\circ$ each.

We are given:

• $m\angle WVY = (4a + 58)^\circ$
• $m\angle XVY = (2b - 18)^\circ$

Since the lines are perpendicular, we have:

$m\angle WVY = 90^\circ \quad \text{and} \quad m\angle XVY = 90^\circ$

Step 1: Solve for $a$

Set $m\angle WVY = 90^\circ$:

$4a + 58 = 90$

Subtract 58 from both sides:

$4a = 32$

Divide by 4:

$a = 8$

Step 2: Solve for $b$

Set $m\angle XVY = 90^\circ$:

$2b - 18 = 90$

Add 18 to both sides:

$2b = 108$

Divide by 2:

$b = 54$

Conclusion:

The values of $a$ and $b$ are $a = 8$ and $b = 54$.

Would you like more details or have any questions about this calculation? Here are some related questions you might find interesting:

1. What would the angles be if $WX$ and $YZ$ were not perpendicular?
2. How do you verify that two lines are perpendicular using their slopes?
3. What is the relationship between complementary and supplementary angles?
4. How would the calculation change if the angles were supplementary instead of right angles?
5. Can you find the coordinates of point $V$ if given the equations of the lines?

Tip: Always double-check your work by substituting the calculated values back into the original equations to ensure that they satisfy the given conditions.