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Theorem \(1\): When two lines intersect, then the vertically opposite angles are equal.
Proof:
Let \(AB\) and \(CD\) be two line segments intersecting at \(O\) as given in the figure.
We should prove that the vertically opposite angles are equal.
The vertically opposite angles are:
1. \(\angle AOD\) and \(\angle BOC\)
2. \(\angle AOC\) and \(\angle BOD\)
Let us consider the vertically opposite angles \(\angle AOD\) and \(\angle BOC\), and prove that they are equal.
\(OD\) is a ray standing on the line \(AB\).
\(\angle AOD + \angle BOD = 180^\circ \longrightarrow (1)\)
[By linear pair of angles axiom 1]
Similarly, \(\angle BOD + \angle BOC = 180^\circ \longrightarrow (2)\)
Let us now equate \((1)\) and \((2)\).
\(\angle AOD + \angle BOD = \angle BOD + \angle BOC\)
Thus, \(\angle AOD = \angle BOC\)
Hence, the vertically opposite angles formed by two intersecting lines are equal.
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