Question

in the figure, $overrightarrow{ba}$ and $overrightarrow{bc}$ are opposite rays and $overrightarrow{bd}$ bisects $angle abe$. if $mangle abd=(2x + 17)^{circ}$ and $mangle dbe=(8x - 28)^{circ}$, find $mangle dbe$. $mangle dbe=square^{circ}$

Explanation:

Step1: Use angle - bisector property

Since $\overrightarrow{BD}$ bisects $\angle ABE$, then $m\angle ABD=m\angle DBE$.
So, $2x + 17=8x-28$.

Step2: Solve the equation for $x$

Subtract $2x$ from both sides: $17 = 8x-2x - 28$, which simplifies to $17=6x - 28$.
Add 28 to both sides: $17 + 28=6x$, so $45 = 6x$.
Divide both sides by 6: $x=\frac{45}{6}=\frac{15}{2}=7.5$.

Step3: Find $m\angle DBE$

Substitute $x = 7.5$ into the expression for $m\angle DBE$.
$m\angle DBE=(8x - 28)^{\circ}$.
$m\angle DBE=8\times7.5-28$.
$m\angle DBE = 60 - 28=32^{\circ}$.

Answer:

$32$