Question

question 7 of 10 given ⊙o below, if ⌢ab and ⌢bc are congruent, what is the measure of ∠boc? a. 50° b. 130° c. 115° d. 65°

Explanation:

Step1: Recall circle - arc relationship

In a circle, congruent arcs subtend equal central angles. Since $\widehat{AB}$ and $\widehat{BC}$ are congruent, $\angle AOB=\angle BOC$.

Step2: Use the fact that the sum of central - angles in a circle is 360°

Let $\angle AOB = \angle BOC=x$. We know that $\angle AOC = 130^{\circ}$. And $\angle AOB+\angle BOC+\angle AOC=360^{\circ}$. But we can also note that the non - reflex $\angle AOC$ and the sum of $\angle AOB$ and $\angle BOC$ make up the whole circle. Since $\angle AOB=\angle BOC$, and the non - reflex $\angle AOC = 130^{\circ}$, and the sum of angles around a point $O$ is 360°, the sum of $\angle AOB$ and $\angle BOC$ is $360^{\circ}- 130^{\circ}=230^{\circ}$.

Step3: Solve for $\angle BOC$

Since $\angle AOB=\angle BOC$ and $\angle AOB+\angle BOC = 230^{\circ}$, then $2\angle BOC=230^{\circ}$. So, $\angle BOC=\frac{230^{\circ}}{2}=115^{\circ}$.

Answer:

C. $115^{\circ}$