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2. Theorem supporting vertically opposite angles

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:

$\angle AOD$ and

$\angle BOC$

$\angle AOC$ 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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