Question

example: in the figure, $overrightarrow{ba}$ and $overrightarrow{bc}$ are opposite rays and $overrightarrow{bd}$ bisects $angle abe$. if $mangle abd=(4x + 14)^{circ}$ and $mangle dbe=(8x - 32)^{circ}$, find $mangle dbe$.

Explanation:

Step1: Use angle - bisector property

Since $\overrightarrow{BD}$ bisects $\angle ABE$, then $m\angle ABD=m\angle DBE$. So we set up the equation $4x + 14=8x-32$.

Step2: Solve the equation for $x$

Subtract $4x$ from both sides: $14 = 8x-4x - 32$, which simplifies to $14=4x - 32$. Then add 32 to both sides: $14 + 32=4x$, so $46 = 4x$. Divide both sides by 4: $x=\frac{46}{4}=\frac{23}{2}=11.5$.

Step3: Find $m\angle DBE$

Substitute $x = 11.5$ into the expression for $m\angle DBE$. $m\angle DBE=(8x - 32)^{\circ}$. So $m\angle DBE=8\times11.5-32$. First, $8\times11.5 = 92$, then $92-32 = 60^{\circ}$.

Answer:

$60^{\circ}$