Question

in the diagram, be bisects ∠dbc. if m∠abc = 144°, what is the measure of ∠dbe? enter your answer in the box. m∠dbe = □

Explanation:

Step1: Set up equation from angle - bisector.

Since $\overline{BE}$ bisects $\angle DBC$, then $\angle DBE=\angle EBC$, so $x = 3x+9$. Solving for $x$:
\[

$$\begin{align*} x-3x&=9\\ - 2x&=9\\ x&=-\frac{9}{2} \end{align*}$$

\]
This is wrong. We should use $\angle ABC$ info. $\angle ABC=x+(3x + 9)=144$.

Step2: Solve for $x$.

\[

$$\begin{align*} x+3x + 9&=144\\ 4x&=144 - 9\\ 4x&=135\\ x&=\frac{135}{4} \end{align*}$$

\]

Step3: Find $\angle DBE$.

Since $\angle DBE = 3x+9$, substitute $x=\frac{135}{4}$ into it.
\[

$$\begin{align*} \angle DBE&=3\times\frac{135}{4}+9\\ &=\frac{405}{4}+9\\ &=\frac{405 + 36}{4}\\ &=\frac{441}{4}=110.25 \end{align*}$$

\]

Answer:

$110.25$