Question

in each diagram, bd bisects ∠abc. find m∠abc. (7x - 6)° (5x + 2)°

Explanation:

Step1: Use angle - bisector property

Since $\overrightarrow{BD}$ bisects $\angle ABC$, then $\angle ABD=\angle CBD$. So, $7x - 6=5x + 2$.

Step2: Solve the equation for $x$

Subtract $5x$ from both sides: $7x-5x - 6=5x-5x + 2$, which simplifies to $2x-6 = 2$. Then add 6 to both sides: $2x-6 + 6=2 + 6$, getting $2x=8$. Divide both sides by 2: $x = 4$.

Step3: Find $\angle ABD$ or $\angle CBD$

Substitute $x = 4$ into the expression for $\angle ABD$ (we could also use $\angle CBD$). $\angle ABD=5x + 2=5\times4+2=20 + 2=22^{\circ}$.

Step4: Find $\angle ABC$

Since $\angle ABC=\angle ABD+\angle CBD$ and $\angle ABD=\angle CBD$, then $\angle ABC = 2\angle ABD$. So, $\angle ABC=2\times22^{\circ}=44^{\circ}$.

Answer:

$44^{\circ}$