Confusion about conformality of Möbius maps

Question

I am aware that Möbius maps are conformal maps, that is, they preserve oriented angles. So I was thinking, say I have a circle in $C$. Then I can find a Möbius map that maps it to a a line (a circle through infinity). Clearly the angle between any two points on the line is the same, however, this is not true for the circle. But Möbius maps are conformal so this must be the case. This seems contradictory. Could someone elaborate on the conformity of Möbius maps that send circles to lines and vice versa?

Answer 1

you can only measure an angle when you have two curves intersecting at some point. Since you are only talking about one curve there is no angle to be found anywhere. If you want to add lines to one of the pictures then you need to also add their images (or inverse images) by the Mobius transform to the other.