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 \(\mathtt{m\color{purple}{\measuredangle{CEA}}}\) + \(\mathtt{m\color{green}{\measuredangle{AED}}}\) = \(\mathtt{\color{blue}{180^\circ}}\).

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.

because those angles also form a linear pair.

3

This means that:

\(\mathtt{m\color{purple}{\measuredangle{CEA}}}\) + \(\mathtt{m\color{green}{\measuredangle{AED}}}\) = \(\mathtt{m\color{orange}{\measuredangle{DEB}}}\) + \(\mathtt{m\color{green}{\measuredangle{AED}}}\).

\(\mathtt{m\color{purple}{\measuredangle{CEA}}}\) + \(\mathtt{m\color{green}{\measuredangle{AED}}}\) = \(\mathtt{m\color{orange}{\measuredangle{DEB}}}\) + \(\mathtt{m\color{green}{\measuredangle{AED}}}\).

4

If \(\mathtt{\color{purple}{a} + \color{green}{b} = \color{orange}{c} + \color{green}{b}}\), then \(\mathtt{\color{purple}{a} = \color{orange}{c}}\).

So, \(\mathtt{m\measuredangle{CEA}}\) = \(\mathtt{m\measuredangle{DEB}}\).

So, \(\mathtt{m\measuredangle{CEA}}\) = \(\mathtt{m\measuredangle{DEB}}\).

5

The same process can be used to show that

\(\mathtt{m\measuredangle{AED}}\) = \(\mathtt{m\measuredangle{BEC}}\).

\(\mathtt{m\measuredangle{AED}}\) = \(\mathtt{m\measuredangle{BEC}}\).

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