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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