Question

bd bisects ∠abc. find ( mangle abd ), ( mangle cbd ), and ( mangle abc ).
(3( x ) + 6)( ^circ )
(7( x ) − 18)( ^circ )
( mangle abd = square^circ )
( mangle cbd = square^circ )
( mangle abc = square^circ )

Explanation:

Step1: Set angles equal (bisector)

Since \( \overrightarrow{BD} \) bisects \( \angle ABC \), \( m\angle ABD = m\angle CBD \). So \( 3x + 6 = 7x - 18 \).

Step2: Solve for \( x \)

Subtract \( 3x \) from both sides: \( 6 = 4x - 18 \).
Add 18 to both sides: \( 24 = 4x \).
Divide by 4: \( x = 6 \).

Step3: Find \( m\angle ABD \)

Substitute \( x = 6 \) into \( 3x + 6 \): \( 3(6) + 6 = 18 + 6 = 24 \).

Step4: Find \( m\angle CBD \)

Since \( m\angle ABD = m\angle CBD \), \( m\angle CBD = 24^\circ \).

Step5: Find \( m\angle ABC \)

\( m\angle ABC = m\angle ABD + m\angle CBD = 24 + 24 = 48 \).

Answer:

\( m\angle ABD = \boldsymbol{24}^\circ \)
\( m\angle CBD = \boldsymbol{24}^\circ \)
\( m\angle ABC = \boldsymbol{48}^\circ \)