Given that $\angle ABD$ and $\angle DBC$ are complementary, this means that the sum of their measures is $90^\circ$. You are provided with the following expressions for the angles:

$m \angle ABD = 5y + 1$ $m \angle DBC = 3y - 7$

Since the angles are complementary, their sum is:

$(5y + 1) + (3y - 7) = 90$

Now, combine like terms:

$5y + 3y + 1 - 7 = 90$ $8y - 6 = 90$

Next, add 6 to both sides:

$8y = 96$

Now, divide both sides by 8:

$y = 12$

Now that we have $y = 12$, we can find the measures of both angles:

$m \angle ABD = 5y + 1 = 5(12) + 1 = 60 + 1 = 61^\circ$ $m \angle DBC = 3y - 7 = 3(12) - 7 = 36 - 7 = 29^\circ$

So, $m \angle ABD = 61^\circ$ and $m \angle DBC = 29^\circ$.

Would you like to explore further details, or do you have any questions?

Here are five related questions to consider:

1. How do we determine if two angles are complementary if their algebraic expressions are more complex?
2. What would happen if the angles were supplementary instead of complementary?
3. Can you solve for $y$ if one of the angles was expressed in terms of $x$ instead of $y$?
4. How would the problem change if $m \angle ABD$ was greater than $90^\circ$?
5. What is the relationship between complementary angles in a right triangle?

Tip: Remember that complementary angles always add up to $90^\circ$, a useful fact when solving problems involving right angles!