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Related Angles - Concept of Vertically Opposite Angles

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definition

Vertically Opposite Angles: When two lines intersect we have two pairs of opposite angles. They are called vertically opposite angles.

Point of intersection: The meeting point of two lines is called the point of intersection.

notes

Vertically Opposite Angles:

  • When two lines intersect we have two pairs of opposite angles. They are called vertically opposite angles.

  • When two lines l and m meet, we say they intersect; the meeting point is called the point of intersection.


  • If two lines intersect each other, then the vertically opposite angles are equal.

  • If two lines intersect at a point, and if one pair of vertically opposite angles are acute angles, then the other pair of vertically opposite angles are Obtuse angles.

  • Adjacent angles are linear and Supplementary angles i.e.,
    ∠AOC + ∠AOD = 180°.
    ∠AOD + ∠DOB = 180°.
    ∠DOB + ∠BOC = 180°.
    ∠BOC + ∠AOC = 180°.

theorem

When two lines intersect, the vertically opposite angles so formed are equal.

To Prove: ∠PQR and ∠SQT are congruent angles.

Proof: 
As shown in the figure, m∠PQS = a, m∠SQT = b, m∠TQR = c, m∠PQR = d.
∠PQS and ∠SQT are the angles in a linear pair.
∴ a + b = 180°

Also, m∠SQT and m∠TQR are two angles in a linear pair.
∴ b + c = 180°
∴ a + b = b + c
∴ a = c.........(Subtracting b from both sides.)
∴ ∠PQS and ∠TQR are congruent angles.

Also, m∠PQR = m∠SQT
i.e. ∠PQR and ∠SQT are congruent angles.

Hence Proved.

Example

In the given figure, Identify two pairs of vertically opposite angles.

Vertically opposite angles are: (∠COB, ∠AOD), and (∠AOC, ∠BOD)

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