The plot is shown in the Figure on the right (although not in the scale).    
The point O is the center of the circle. 
It is connected with the point C by the segment OC.
OA and OB are the radii drawn to the tangent points A and C respectively.
We are given that the measure of the minor arc AC is 110°.
It means that the measure of the central angle AOC is 110°.

Consider the triangles OAB and OCB.
These triangles are right angled triangles.

They have the common hypotenuse OB and congruent legs OA and OC.
Hence, the triangles are congruent.
It implies that the angles AOB and COB are congruent.
Then the measure of each of these angles is the half o the measure 
of the angle AOC, i.e. 55°.

The angle ABO is complement to the angle AOB and, therefore, is 
equal to 90°-55° = 35°.

Similarly, the angle CBO is complement to the angle COB and, therefore, 
is equal to 90°-55° = 35°.
 
  

 

                Figure. 

  
  
Then the angle ABC is 35° + 35° = 70°.

Answer.  Angle ABC is 70°.