Question

→ bd bisects ∠abc. find m∠abd, m∠cbd, and m∠abc.
image: angle with points a, b, c, d; ∠abd is (8x + 35)°, ∠cbd is (11x + 23)°

m∠abd = □°
m∠cbd = □°
m∠abc = □°

Explanation:

Step1: Set angles equal (bisector property)

Since \( \overrightarrow{BD} \) bisects \( \angle ABC \), \( m\angle ABD = m\angle CBD \). So, \( 8x + 35 = 11x + 23 \).

Step2: Solve for \( x \)

Subtract \( 8x \) and \( 23 \) from both sides:
\( 35 - 23 = 11x - 8x \)
\( 12 = 3x \)
Divide by 3: \( x = 4 \).

Step3: Find \( m\angle ABD \)

Substitute \( x = 4 \) into \( 8x + 35 \):
\( 8(4) + 35 = 32 + 35 = 67 \).

Step4: Find \( m\angle CBD \)

Since \( m\angle ABD = m\angle CBD \), \( m\angle CBD = 67^\circ \).

Step5: Find \( m\angle ABC \)

\( m\angle ABC = m\angle ABD + m\angle CBD = 67 + 67 = 134 \).

Answer:

\( m\angle ABD = \boldsymbol{67}^\circ \)
\( m\angle CBD = \boldsymbol{67}^\circ \)
\( m\angle ABC = \boldsymbol{134}^\circ \)