If the blue angle measures MEASURE degrees, what does the orange angle measure?
This is a special case where the center of the circle is outside the inscribed orange angle. The blue angle is called a central angle.
What do we know about the angles formed by the dashed diameter shown above?
From the previous inscribed angles exercises, we know the following about the green and pink angles.
\color{GREEN}{\text{green angle}} = \dfrac{1}{2} \cdot \color{PINK}{\text{pink angle}}
We can see another pair of these special case inscribed and central angles, with the same relationship between green and pink angles.
Looking at the picture, we can see the following is true:
\color{GREEN}{\text{small green angle}} + \color{ORANGE}{\text{orange angle}} = \color{GREEN}{\text{big green angle}}
Substituting what we know about green and pink angles, we get the following:
\dfrac{1}{2} \cdot \color{PINK}{\text{small pink angle}} + \color{ORANGE}{\text{orange angle}} = \dfrac{1}{2} \cdot \color{PINK}{\text{big pink angle}}
\color{ORANGE}{\text{orange angle}} = \dfrac{1}{2}( \color{PINK}{\text{big pink angle}} - \color{PINK}{\text{small pink angle}})
We can see from the picture that the following is also true:
\color{PINK}{\text{small pink angle}} + \color{BLUE}{\text{blue angle}} = \color{PINK}{\text{big pink angle}}
\color{BLUE}{\text{blue angle}} = \color{PINK}{\text{big pink angle}} - \color{PINK}{\text{small pink angle}}
Combining what we know about blue and orange angles:
\color{ORANGE}{\text{orange angle}} = \dfrac{1}{2} \cdot \color{BLUE}{\text{blue angle}}
\color{ORANGE}{\text{orange angle}} = \dfrac{1}{2} \cdot \color{BLUE}{MEASURE^{\circ}}
\color{ORANGE}{\text{orange angle}} = \color{BLUE}{MEASURE / 2^{\circ}}