Inscribed Angles and Polygons. Theorem 10.8

Question

How to find an indicated arc using the theorem " if two inscribed angles of a circle intercept the same arc, then the angles are congruent.

Answer 1

 

∠ADO=∠DAO=x, ∠ODB=∠DBO=y (isosceles ∆s AOD and BOD).

∠AOD=180-2x, ∠BOD=180-2y, ∠AOB=360-(180-2x+180-2y),

∠AOB=2(x+y). But x+y=∠ADB.

∠ACO=∠CAO=w, ∠OCB=∠OBC=z (isosceles ∆s COA and COB).

∠AOC=180-2w, ∠BOC=180-2z,

∠AOB=∠BOC-∠AOC=180-2z-(180-2w)=2(w-z).

But w-z=∠ACB.

Therefore ∠AOB=2(∠ADB)=2(∠ACB) and ∠ADB=∠ACB, that is, 

Angles at the circumference of a circle subtended by the same chord or arc are congruent.