Question

ding an angle measure
what is the measure of angle aoc?
○ 42°
○ 58°
○ 66°
○ 79°
58°
50°

Explanation:

Step1: Find ∠DAB (inscribed angle)

$\angle DAB = \frac{1}{2} \times 50^\circ = 25^\circ$

Step2: Find ∠AOC (exterior angle)

$\angle AOC = \angle DAB + \angle ABC = 25^\circ + 58^\circ = 83^\circ$
*Note: Correcting using arc sum: Arc DE = 50°, Arc DC = 2×58°=116°, so Arc AC = 180° - (116°-50°)=114°? No, correct inscribed angle for Arc AC: ∠ABC is exterior to △ABO, ∠AOC is central angle for Arc AC. Arc AE = 2×∠ADE? No, correct: ∠DBC=58° is inscribed over Arc DC, so Arc DC=116°, Arc DE=50°, so Arc EC=116°-50°=66°, Arc AC=180°-66°=114°? No, central angle ∠AOC is half of Arc AC? No, central angle equals arc measure. Wait, ∠ABE=58°, vertical angle ∠ABD=58°, inscribed angle over Arc AD, so Arc AD=2×58°=116°, Arc DE=50°, so Arc AE=180°-116°-50°=14°? No, correct: ∠DCE=58° is not, the 58° is ∠DBA, which is inscribed over Arc AD, so Arc AD=116°, Arc DE=50°, so Arc AE=360°-180°-50°? No, AD is diameter? O is center, so AD is diameter, Arc AD=180°. Then Arc DE=50°, so Arc AE=180°-50°=130°? No, ∠DBE is 58°, which is inscribed over Arc DE? No, ∠DBC=58°, which is inscribed over Arc DC, so Arc DC=116°, Arc DE=50°, so Arc EC=116°-50°=66°, so Arc AC=180°-66°=114°? No, central angle ∠AOC equals Arc AC, but options don't have 114. Wait, ∠OBC=58°, OB=OC, so ∠OCB=∠OBC=58°, ∠BOC=180-2×58=64°, ∠AOB=∠DBE=58° (vertical), so ∠AOC=58+64=122? No. Wait, correct: ∠ADE=50° is inscribed over Arc AE, so Arc AE=100°, ∠DBA=58° is inscribed over Arc AD, but AD is diameter, Arc AD=180°, so Arc AE=100°, Arc ED=80? No, the 50° is Arc DE, so Arc DE=50°, inscribed angle over Arc DE is 25°, which is ∠DAE. ∠ABD=58°, so in △ABD, ∠ADB=180-58-25=97? No. Final correct: ∠AOC is equal to 2×(∠ABC - ∠BAE). ∠BAE=25°, ∠ABC=58°, so ∠AOC=2×(58-25)=66°

Step2 (Corrected): Calculate ∠AOC

$\angle AOC = 2\times(58^\circ - 25^\circ) = 66^\circ$

Answer:

66° (Option C)