Question

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

Explanation:

Step1: Find measure of minor arc AB

The sum of major and minor arcs of a circle is 360°. Given major arc AB = 220°, so minor arc AB = 360° - 220° = 140°.

Step2: Find central - angle AOB

The measure of a central - angle is equal to the measure of its intercepted arc. So, ∠AOB=140°.

Step3: Consider triangle AOC

Since OA = OC (radii of the same circle), triangle AOC is isosceles. Let ∠OAC = ∠OCA = x.

Step4: Use angle - sum property of a triangle in triangle AOC

The sum of interior angles of a triangle is 180°. In triangle AOC, we know that ∠AOC is half of the central - angle corresponding to minor arc AB (the central - angle theorem for inscribed angles). The central - angle corresponding to minor arc AB is 140°, so ∠AOC = 70°.
Using the angle - sum property in △AOC: ∠OAC+∠OCA + ∠AOC=180°. Since ∠OAC = ∠OCA = x and ∠AOC = 70°, we have 2x+70° = 180°.
2x=180° - 70° = 110°.
x = 55°.

Answer:

55°