abd = 47, dbc = 43
The fact that the rays ab and bc are perpendicular tells you that the angle abc is 90 degrees. Given that the point d is within the interior of the angle abc tells you several things. Namely that the angle abd is less than or equal to 90, the angle dbc is also less than or equal to 90, and finally that the sum of the angles abd and dbc total to exactly 90. So write the formula
abd + dbc = 90
Substitute the formulas given for the angles
(3r + 5) + (5r - 27) = 90
Get rid of the parenthesis
3r + 5 + 5r - 27 = 90
Merge the terms
8r - 22 = 90
Add 22 to both sides
8r = 112
Divide both sides by 8
r = 14
Now given r = 14, calculate both abd and dbc
abd = 3r + 5 = 3*14 + 5 = 42 + 5 = 47
dbc = 5r - 27 = 5*14 - 27 = 70 - 27 = 43
