# Pentagram in a Circle +-- {.image} [[PentagraminaCircle.png:pic]] > Angles of the same colour are the same size. What's the orange angle? =-- ## Solution by [[Angles in a Semi-Circle]], [[Angles in the Same Segment]], and [[Angles in a Triangle]] +-- {.image} [[PentagraminaCircleLabelled.png:pic]] =-- 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 $180^\circ - 30^\circ - 45^\circ = 105^\circ$.