Question

if $overrightarrow{ef}$ bisects $angle ceb$, $mangle cef=(7x + 21)^{circ}$ and $mangle feb=(10x - 3)^{circ}$, find the measure of $angle deb$.

Explanation:

Step1: Use angle - bisector property

Since $\overrightarrow{EF}$ bisects $\angle CEB$, then $m\angle CEF=m\angle FEB$. So we set up the equation $7x + 21=10x-3$.

Step2: Solve the equation for $x$

Subtract $7x$ from both sides: $21 = 3x-3$. Then add 3 to both sides: $24=3x$. Divide both sides by 3, we get $x = 8$.

Step3: Find the measure of $\angle FEB$

Substitute $x = 8$ into the expression for $m\angle FEB$. So $m\angle FEB=(10\times8 - 3)^{\circ}=77^{\circ}$.

Step4: Assume $\angle DEB$ and $\angle CEA$ are vertical angles

We need more information about the relationship between $\angle DEB$ and the other angles. But if we assume that $\angle CEA$ and $\angle DEB$ are vertical - angles and $\angle CEB$ is a straight - angle ($180^{\circ}$), and since $\angle CEF=\angle FEB = 77^{\circ}$, then $\angle CEA=180-(77 + 77)=26^{\circ}$. So $\angle DEB = 26^{\circ}$.

Answer:

$26^{\circ}$