Question

determining unknown angle measures
what is the measure of angle oac if major arc ab measures 220 degrees?
○ 55°
○ 70°
○ 110°
○ 140°

Explanation:

Step1: Find minor arc AB

The total degrees in a circle is $360^\circ$. Subtract major arc AB from $360^\circ$.
$\text{Minor arc } AB = 360^\circ - 220^\circ = 140^\circ$

Step2: Find $\angle AOB$

The central angle equals its arc measure.
$\angle AOB = 140^\circ$

Step3: Find $\angle OAB$

In isosceles $\triangle OAB$ ($OA=OB$, radii), sum of angles is $180^\circ$.
$\angle OAB = \frac{180^\circ - 140^\circ}{2} = 20^\circ$

Step4: Find $\angle ACB$

Inscribed angle is half its arc measure.
$\angle ACB = \frac{1}{2} \times 220^\circ = 110^\circ$

Step5: Find $\angle CAB$

In isosceles $\triangle ACB$ ($AC=BC$), sum of angles is $180^\circ$.
$\angle CAB = \frac{180^\circ - 110^\circ}{2} = 35^\circ$

Step6: Calculate $\angle OAC$

Add $\angle OAB$ and $\angle CAB$.
$\angle OAC = 20^\circ + 35^\circ = 55^\circ$

Answer:

55° (Option A)