Question

rays ba and bc are perpendicular. point d lies in the interior of ∠abc. if ( mangle abd = (3x + 5)^circ ) and ( mangle dbc = (5x - 27)^circ ), find ( mangle abd ) and ( mangle dbc ). fill in the blanks below with the correct responses for the value of ( x ), and the two angle measures.
( x = )
( mangle abd = )
( mangle dbc = )

Explanation:

Step1: Use perpendicular angles sum to 90°

Since \( BA \perp BC \), \( \angle ABC = 90^\circ \). And \( \angle ABD + \angle DBC=\angle ABC \), so \( (3x + 5)+(5x - 27)=90 \).

Step2: Solve for x

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

Step3: Find \( m\angle ABD \)

Substitute \( x = 14 \) into \( 3x + 5 \): \( 3(14)+5=42 + 5 = 47^\circ \).

Step4: Find \( m\angle DBC \)

Substitute \( x = 14 \) into \( 5x - 27 \): \( 5(14)-27=70 - 27 = 43^\circ \).

Answer:

\( x = 14 \)
\( m\angle ABD = 47^\circ \)
\( m\angle DBC = 43^\circ \)