Vertical Angles:

When two lines meet at a point in a plane, they are known as intersecting lines. When the lines do not meet at any point in a plane, they are called parallel lines. The given figure shows intersecting and parallel lines.

In the figure given above, the line segment \(\overline{AB}\) and \(\overline{CD}\) meet at the point \(O\) and these represent two intersecting lines. The line segment \(\overline{PQ}\) and \(\overline{RS}\) represent two parallel lines as they have no common intersection point in the given plane. Infinite lines can pass through a single point and hence through \(O\), multiple intersecting lines can be drawn. Similarly, infinite parallel lines can be drawn parallel to \(\overline{PQ}\) and \(\overline{RS}\) . Also, it is worth noting that the perpendicular distance between two parallel lines is constant.

In a pair of intersecting lines, the angles which are opposite to each other form a pair of vertically opposite angles. In the figure given above, âˆ AOD and âˆ COB form a pair of vertically opposite angle and similarly âˆ AOC and âˆ BOD form such a pair. These angles are also known as vertical angles or opposite angles.

For a pair of opposite angles the following theorem, known as vertical angle theorem holds true:

Theorem: In a pair of intersecting lines the vertically opposite angles are equal.

Proof: Consider two lines \(\overleftrightarrow{AB}\) and \(\overleftrightarrow{CD}\) which intersect each other at \(O\). The two pairs of vertical angles are: i) âˆ AOD and âˆ COB ii) âˆ AOC and âˆ BOD as shown.

It can be seen that ray \(\overline{OA}\) stands on the line \(\overleftrightarrow{CD}\) and according to Linear Pair Axiom, if a ray stands on a line, then the adjacent angles form a linear pair of angles.

Therefore, âˆ AOD + âˆ AOC = 180Â° —(1) (Linear pair of angles)

Similarly, \(\overline{OC}\) stands on the line \(\overleftrightarrow{AB}\).

Therefore, âˆ AOC + âˆ BOC = 180Â° —(2) (Linear pair of angles)

From (1) and (2),

âˆ AOD + âˆ AOC = âˆ AOC + âˆ BOC

â‡’ âˆ AOD = âˆ BOC —(3)

Also, \(\overline{OD}\) stands on the line \(\overleftrightarrow{AB}\)..

Therefore, âˆ AOD + âˆ BOD = 180Â° —(4) (Linear pair of angles)

From (1) and (4),

âˆ AOD + âˆ AOC = âˆ AOD + âˆ BOD

â‡’ âˆ AOC = âˆ BOD —(5)

Thus, the pair of opposite angles are equal.

Consider the figure given below to understand this concept.

In the given figure âˆ AOC = âˆ BOD and âˆ COB = âˆ AOD(Vertical Angles)

â‡’ âˆ BOD = 105Â° and âˆ AOD = 75Â°

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