The cutting angles are automatically right if your shape is a circle, you only split it, remove the unwanted pieces and give the wanted stroke thickness:

[![enter image description here][1]][1]

In the middle some new anchor points are unserted, the circle is splitted at them using the Scissors tool and the unwanted pieces are deleted.

In the right a new wide stroke has been given.

Object > Path > Outline stroke converts the pieces to fillable areas which can have new strokes:

[![enter image description here][2]][2]

Warning: Partially deleted circle is difficult to move, scale and rotate, because the original center is easily lost. Keep it grouped with a bigger solid circle or square until the placement and other transformations are fixed. Learn how to use the layers panel to manage different parts.


  [1]: https://i.sstatic.net/45twI.jpg
  [2]: https://i.sstatic.net/HCJhL.jpg