Question

Find the value of b.

Answer 1

Answer: $230^{\circ$

Step-by-step explanation:

For convenience, I labeled some points as shown in the attached picture.

Also, I assume $\overline{AB}$ and $\overline{BC}$ are tangents to the circle.

1. $\overline{BO} \cong \overline{BO}$ (reflexive property)
2. $\overline{AB} \cong \overline{BC}$ (tangents drawn from a common external point are congruent)
3. $\angle BAO \cong \angle BCO$ (right angles are congruent)

Therefore, we know $\triangle ABO \cong \triangle CBO$ by HL.

Thus, by CPCTC,

$\angle AOB \cong \angle BOC \implies m\angle BOC=65^{\circ}\\\\\implies m\angle AOC=130^{\circ}$

This means the measure of minor arc AC is $130^{\circ}$, and thus $b=360^{\circ}-130^{\circ}=\boxed{230^{\circ}}$