Question

bd bisects ∠abc. find m∠abd, m∠cbd, and m∠abc. m∠abc=(2 - 16x)°. m∠abd=□°. m∠cbd=□°. m∠abc=□°

Explanation:

Step1: Use angle - bisector property

Since $\overrightarrow{BD}$ bisects $\angle ABC$, then $m\angle ABD=m\angle CBD$ and $m\angle ABC = 2m\angle ABD=2m\angle CBD$. So we set up the equation $2x + 81=\frac{1}{2}(2 - 16x)$.

Step2: Solve the equation for $x$

First, multiply both sides of the equation $2x + 81=\frac{1}{2}(2 - 16x)$ by 2 to get $4x+162 = 2-16x$. Then add $16x$ to both sides: $4x + 16x+162=2-16x+16x$, which simplifies to $20x+162 = 2$. Next, subtract 162 from both sides: $20x+162 - 162=2 - 162$, giving $20x=-160$. Divide both sides by 20: $x=\frac{-160}{20}=-8$.

Step3: Find $m\angle ABD$

Substitute $x = - 8$ into the expression for $m\angle ABD=2x + 81$. Then $m\angle ABD=2(-8)+81=-16 + 81 = 65^{\circ}$.

Step4: Find $m\angle CBD$

Since $m\angle ABD=m\angle CBD$, then $m\angle CBD = 65^{\circ}$.

Step5: Find $m\angle ABC$

$m\angle ABC=2m\angle ABD$, so $m\angle ABC = 2\times65^{\circ}=130^{\circ}$.

Answer:

$m\angle ABD = 65^{\circ}$
$m\angle CBD = 65^{\circ}$
$m\angle ABC = 130^{\circ}$