Question

rays ba and bc are perpendicular. point d lies in the interior of ∠abc. if m∠abd = (3r + 5)° and m∠dbc = (5r − 27)°, find m∠abd and m∠dbc.
m∠abd = \boxed{\space}°
m∠dbc = \boxed{\space}°

Explanation:

Step1: Determine the measure of ∠ABC

Since rays \( BA \) and \( BC \) are perpendicular, \( \angle ABC = 90^\circ \). Also, \( \angle ABD + \angle DBC=\angle ABC \), so we have the equation \((3r + 5)+(5r - 27)=90\).

Step2: Solve for \( r \)

Combine like terms: \( 3r+5r + 5-27 = 90 \), which simplifies to \( 8r-22 = 90 \).
Add 22 to both sides: \( 8r=90 + 22=112 \).
Divide both sides by 8: \( r=\frac{112}{8} = 14 \).

Step3: Find \( m\angle ABD \)

Substitute \( r = 14 \) into \( m\angle ABD=(3r + 5)^\circ \): \( 3\times14+5=42 + 5=47^\circ \).

Step4: Find \( m\angle DBC \)

Substitute \( r = 14 \) into \( m\angle DBC=(5r - 27)^\circ \): \( 5\times14-27=70 - 27 = 43^\circ \).

Answer:

\( m\angle ABD = 47^\circ \)
\( m\angle DBC = 43^\circ \)