Rays ab and bc are perpendicular. point d lies in the interior of angleabc. if angleabd=3r+5 and angledbc=5r-27, find angleabd and angledbc.

The first step is to always illustrate the problem as shown in the attached picture. Lines AB and BC are perpendicular, meaning they for right angles with each other. Then, draw a random point D within angle ABC, then connect it towards vertex B. Two angles are formed with the following measurements in terms of r. To solve this problem, we apply Angle Addition Postulate. It states that the included angles within the whole angle, could sum up to the whole angle. Therefore, ∠ABD and ∠DBC should sum up to 90°.

∠ABC = ∠ABD + ∠DBC
90° = (3r + 5) + (5r - 27)
90 = 3r + 5 + 5r - 27
90 = 8r -22
8r = 112
r = 112/8
r = 14

Knowing r, we could substitute this to individual angle measurements given.

∠ABD = 3(14) + 5
∠ABD = 47°

∠DBC = 5(14) - 27
∠DBC = 43°