vertical angle theorem vertical angle theorem
Vertical Angle Theorem
Press on the numbers to see the steps of the proof.
1
\(\mathtt{\overleftrightarrow{AB}}\) and \(\mathtt{\overleftrightarrow{CD}}\) intersect at point \(\mathtt{E}\).
And \(\mathtt{m\color{purple}{\measuredangle{CEA}}}\) + \(\mathtt{m\color{green}{\measuredangle{AED}}}\) = \(\mathtt{\color{blue}{180^\circ}}\).
2
Similarly, \(\mathtt{m\color{green}{\measuredangle{AED}}}\) + \(\mathtt{m\color{orange}{\measuredangle{DEB}}}\) = \(\mathtt{\color{blue}{180^\circ}}\),
because those angles also form a linear pair.
3
This means that:
\(\mathtt{m\color{purple}{\measuredangle{CEA}}}\) + \(\mathtt{m\color{green}{\measuredangle{AED}}}\) = \(\mathtt{m\color{orange}{\measuredangle{DEB}}}\) + \(\mathtt{m\color{green}{\measuredangle{AED}}}\).
4
If \(\mathtt{\color{purple}{a} + \color{green}{b} = \color{orange}{c} + \color{green}{b}}\), then \(\mathtt{\color{purple}{a} = \color{orange}{c}}\).
So, \(\mathtt{m\measuredangle{CEA}}\) = \(\mathtt{m\measuredangle{DEB}}\).
5
The same process can be used to show that
\(\mathtt{m\measuredangle{AED}}\) = \(\mathtt{m\measuredangle{BEC}}\).
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