Question

angle bisector and inverse hw
$overrightarrow{bx}$ bisects $angle abc$
if $angle abx = 4z - 16$ and $angle cbx=2z + 6$
z =
$angle abc=$

Explanation:

Step1: Use angle - bisector property

Since $\overrightarrow{BX}$ bisects $\angle ABC$, then $\angle ABX=\angle CBX$. So we set up the equation $4z - 16=2z + 6$.

Step2: Solve the equation for $z$

Subtract $2z$ from both sides: $4z-2z - 16=2z-2z + 6$, which simplifies to $2z-16 = 6$. Then add 16 to both sides: $2z-16 + 16=6 + 16$, getting $2z=22$. Divide both sides by 2: $z=\frac{22}{2}=11$.

Step3: Find $\angle ABC$

First find $\angle ABX$ or $\angle CBX$ by substituting $z = 11$ into either expression. Using $\angle ABX=4z - 16$, we have $\angle ABX=4\times11-16=44 - 16 = 28$. Since $\angle ABC = 2\angle ABX$ (because of angle - bisector), then $\angle ABC=2\times28 = 56$.

Answer:

$z = 11$
$\angle ABC=56$