Notes
pentagram in a circle solution

Pentagram in a Circle

Solution by Angles in a Semi-Circle, Angles in the Same Segment, and Angles in a Triangle

With the points labelled as in the diagram, the line segment $A C$ is a diameter of the circle. Since the angle in a semi-circle is a right-angle, angles $A \hat{E} C$, $A \hat{D} C$, and $C \hat{B} A$ are all $90^\circ$. Then as angles in the same segment are equal, angle $A \hat{E} B$ is the same as angle $A \hat{D} B$. This means that the red angle is $45^\circ$. Also, angles $D \hat{A} E$ and $D \hat{B} E$ are the same so as angles in a triangle add up to $180^\circ$, considering triangle $A E C$ shows that the blue angle is $30^\circ$.

Again, using angles in the same segment then angle $A \hat{C} B$ is the same as $A \hat{D} B$, which is $45^\circ$, and angle $D \hat{B} C$ is the same as $D \hat{A} C$, which is $30^\circ$. So as angles in a triangle add up to $180^\circ$, angle $B \hat{F} C$ is $180^\circ - 30^\circ - 45^\circ = 105^\circ$.