+167 Points $A$ and $B$ are on circle $O$ such that arc $AB$ is 80 degrees. A circle is constructed that passes through $A$, $B$, and $O$. Find the measure of arc $AOB$ on this circle.
+128460
Draw OA , OB and AB
Now since minor arc AB = 80°....then, in triangle AOB, since angle AOB is a central angle of the larger circle subtending minor arc AB, it has the same measure.
And OA = OB since they are both radii of the larger circle....but if OA = OB....then the angles opposite these sides in triangle AOB - namely angles OAB and OBA - are equal....
Thus.... OAB = OBA = [ 180 - 80 ] / 2 = 50°
But angles OAB and OBA are inscribed angles in the smaller circle
And OAB intercepts minor arc OB.....and its measure is twice that of OAB =100°
Likewise....OBA intercepts minor arc OA....and its measure is twice that of OBA = 100°
Thus minor arc OA + minor arc OB = arc AOB
And 100° + 100° = arc AOB = 200°
