Question

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.

Answer 1

To find the measures of angle ABD and angle DBC, we can use the fact that rays AB and BC are perpendicular and the given angle measures. The measures of angle ABD and angle DBC are 47 degrees and 43 degrees, respectively.

Step-by-step explanation:

To find the measures of angle ABD and angle DBC, we can use the fact that rays AB and BC are perpendicular and the given angle measures. Since rays AB and BC are perpendicular, angle ABD and angle DBC are complementary, meaning they add up to 90 degrees.

We know that angle ABD = 3r + 5 and angle DBC = 5r - 27. Substituting these values into the equation ABD + DBC = 90, we get (3r + 5) + (5r - 27) = 90.

Simplifying the equation, we have 8r - 22 = 90. Solving for r, we find r = 14. Plugging this value back into the equations for angle ABD and angle DBC, we get angle ABD = 3(14) + 5 = 47 degrees and angle DBC = 5(14) - 27 = 43 degrees.

Answer 2

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°