Prove that vertically oppossite angles are equal.

The straight lines AB and CD intersect in O.
∴ ∠AOC + ∠AOD = 180° … (1) (Linear pair)
∠AOD + ∠BOD = 180° … (2) (Linear pair)
From (1) and (2), we get
∠AOC + ∠AOD = ∠AOD + ∠BOD
Similarly, ∠BOC = ∠AOD
Thus, if two straight intersect each other, then the pair of vertically opposite angles are equal.


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    Theorem: All vertically opposite angles have equal measure.

      (To get started, we first use the definition of vertically opposite angles to make sense of the statement. We sketch a labeled figure to introduce notation. We then restate what must be shown using the explicit notation.)


      Let one pair of vertically opposite angles in the following figure have measures s and t and the other have measures u and v.

      We must show s=t and u=v.

      s+u=straight angle=t+u. Therefore s=t.

      Likewise, s+u=straight angle=s+v. Therefore u=v.

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