Rays AB and BC are perpendicular. Point D lies in the interior of angle ABC. If Angle ABD = 4r-7 and angle DCB = 8r+1, find angles ABC and DCB.

To find the measures of angles ABC and DCB, we need to use the fact that rays AB and BC are perpendicular.

First, let's find the relationship between angles ABD and DCB. Since AB and BC are perpendicular, we can use the property that the sum of angles in a right triangle is always 90 degrees.

Therefore, we have:

Angle ABD + Angle DCB = 90 degrees

Substituting the given values, we have:

(4r - 7) + (8r + 1) = 90

Combining like terms, we get:

12r - 6 = 90

Adding 6 to both sides:

12r = 96

Dividing both sides by 12:

r = 8

Now that we have found the value of r, we can substitute it back into the expressions for angles ABD and DCB to find their measures.

Angle ABD = 4r - 7
Angle ABD = 4(8) - 7
Angle ABD = 32 - 7
Angle ABD = 25 degrees

Angle DCB = 8r + 1
Angle DCB = 8(8) + 1
Angle DCB = 64 + 1
Angle DCB = 65 degrees

Therefore, the measures of angles ABC and DCB are 25 degrees and 65 degrees, respectively.