Question

given
\\(\overline{oa}\perp\overline{oc}\\)
\\(m\angle boc = 6x - 6^{circ}\\)
\\(m\angle aob = 5x + 8^{circ}\\)
find (m\angle boc):

Explanation:

Step1: Use perpendicular - angle property

Since $\overline{OA}\perp\overline{OC}$, then $\angle AOC = 90^{\circ}$, and $\angle AOB+\angle BOC=\angle AOC = 90^{\circ}$.

Step2: Substitute angle expressions

Substitute $m\angle BOC = 6x - 6^{\circ}$ and $m\angle AOB = 5x + 8^{\circ}$ into $\angle AOB+\angle BOC = 90^{\circ}$, we get $(5x + 8)+(6x - 6)=90$.

Step3: Simplify the equation

Combine like - terms: $5x+6x+8 - 6=90$, which simplifies to $11x+2 = 90$.

Step4: Solve for $x$

Subtract 2 from both sides: $11x=90 - 2=88$. Then divide both sides by 11, so $x = 8$.

Step5: Find $m\angle BOC$

Substitute $x = 8$ into the expression for $m\angle BOC$: $m\angle BOC=6x - 6$. So $m\angle BOC=6\times8 - 6=48 - 6 = 42^{\circ}$.

Answer:

$42^{\circ}$