Interior Angles of a Circle Theorem

Press on the numbers to see the steps of the proof.

1

Draw Circle \(\mathtt{A}\) and chords \(\mathtt{\color{green}{\overline{CD}}}\) and \(\mathtt{\color{blue}{\overline{EF}}}\) which intersect inside the circle at point \(\mathtt{G}\).

2

Connect points \(\mathtt{C}\) and \(\mathtt{F}\) to create a line segment, \(\mathtt{\overline{CF}}\). This forms \(\mathtt{\color{orange}{\Delta FGC}}\).

3

By the Inscribed Angle Theorem, \(\mathtt{\color{DeepPink}{m\angle{FCD}}}\) is \(\mathtt{\frac{1}{2}}\)(arc \(\mathtt{\color{red}{FD}}\)). And \(\mathtt{\color{DeepSkyBlue}{m\angle{EFC}}}\) is \(\mathtt{\frac{1}{2}}\)(arc \(\mathtt{\color{red}{CE}}\)).

4

\(\mathtt{\color{purple}{\angle{FGD}}}\) is an exterior angle of \(\mathtt{\color{orange}{\Delta FGC}}\). By the Exterior Angle Theorem, \(\mathtt{\color{purple}{m\angle{FGD}} = \color{DeepPink}{m\angle{FCD}} + \color{DeepSkyBlue}{m\angle{EFC}}}\).

5

The measure of \(\mathtt{\color{purple}{\angle{FGD}}}\) is equal to \(\mathtt{\frac{1}{2}}\)(arc \(\mathtt{\color{red}{FD}}\) + arc \(\mathtt{\color{red}{CE}}\)).

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