На рисунке 84 ∠AOD + ∠AOC + ∠COB = 210°. Найдите углы AOD и DOB.

Solution:
1. ∠AOD + ∠AOC + ∠COB = 210° 2. ∠AOD + ∠DOB = 180° (angles on a straigh line) 3. ∠AOC = ∠DOB as vertical angles. 4. ∠AOD + ∠AOC + ∠COB = 210° can be written as ∠AOD + ∠DOB + ∠COB = 210° 5. From point 2 we know that ∠AOD + ∠DOB = 180°. 6. ∠COB = 210° - 180° = 30° 7. ∠AOC = ∠DOB as vertical angles. ∠AOD + ∠AOC + ∠COB = 210° can be written as ∠AOD + ∠DOB + ∠COB = 210° then as ∠AOD + ∠DOB = 210° - ∠COB 8. Now we know ∠COB = 30°, so ∠AOD + ∠DOB = 210° - 30° = 180° 9. So ∠DOB = 180° - ∠AOD 10. ∠AOD + ∠AOC + ∠COB = 210°, ∠AOC = ∠DOB . ∠AOD +∠DOB=210 -∠COB 11. ∠COB = 30, ∠AOD+∠DOB=180 12. ∠AOD=x, ∠DOB=y x+y+30=210; x+y=180 then x=180-y (180-y)+y+30=210; y+y=180; then y=90; then x = 90
Answer: ∠AOD = 90°, ∠DOB = 90°