Prove that if two lines intersect then the vertically opposite angles are equal Class 9

From the figure, we know that,

AB and CD intersect each other at point O.

Let the two pairs of vertically opposite angles be,

1st pair - ∠AOC and ∠BOD

2nd pair - ∠AOD and ∠BOC

To prove:

Vertically opposite angles are equal,

i.e., ∠AOC = ∠BOD, and ∠AOD = ∠BOC

From the figure,

The ray AO stands on the line CD.

We know that,

If a ray lies on a line then the sum of the adjacent angles is equal to 180°.

⇒ ∠AOC + ∠AOD = 180° (By linear pair axiom) … (i)

Similarly, the ray DO lies on the line AOB.

⇒ ∠AOD + ∠BOD = 180° (By linear pair axiom) … (ii)

From equations (i) and (ii),

We have,

∠AOC + ∠AOD = ∠AOD + ∠BOD

⇒ ∠AOC = ∠BOD - - - - (iii)

Similarly, the ray BO lies on the line COD.

⇒ ∠DOB + ∠COB = 180° (By linear pair axiom) - - - - (iv)

Also, the ray CO lies on the line AOB.

⇒ ∠COB + ∠AOC = 180° (By linear pair axiom) - - - - (v)

From equations (iv) and (v),

We have,

∠DOB + ∠COB = ∠COB + ∠AOC

⇒ ∠DOB = ∠AOC - - - - (vi)

Thus, from equation (iii) and equation (vi),

We have,

∠AOC = ∠BOD, and ∠DOB = ∠AOC

Therefore, we get, vertically opposite angles are equal.

Hence Proved.

Question 1 If two lines intersect, prove that the vertically opposite angles are equal.

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Answer

Verified

We know that the sum of angles lying on a straight line is ${{180}^{\circ }}$.Let us consider the angles on the line CD. So, we get $\angle COB+\angle BOD={{180}^{\circ }}$.$\Rightarrow \angle COB={{180}^{\circ }}-\angle BOD$ ---(1).Now, let us consider the angles on the line AB. So, we get $\angle COB+\angle AOC={{180}^{\circ }}$ ---(2).Let us substitute equation (2) in equation (1).So, we get ${{180}^{\circ }}-\angle BOD+\angle AOC={{180}^{\circ }}$.$\Rightarrow \angle AOC={{180}^{\circ }}-{{180}^{\circ }}+\angle BOD$.$\Rightarrow \angle AOC=\angle BOD$.From the figure, we can see that the angles $\angle AOC$ and $\angle BOD$ are vertically opposite angles.So, we have proved that if two lines intersect each other, then the vertically opposite angles are equal.

Note: We can also prove that the other pair of vertically opposite angles $\angle COB$ and $\angle DOA$ equal in the similar way as shown below:

Let us consider the angles on the line CD. So, we get $\angle COB+\angle BOD={{180}^{\circ }}$.$\Rightarrow \angle BOD={{180}^{\circ }}-\angle COB$ ---(3).Now, let us consider the angles on the line AB. So, we get $\angle BOD+\angle DOA={{180}^{\circ }}$ ---(4).Let us substitute equation (3) in equation (4).So, we get ${{180}^{\circ }}-\angle COB+\angle DOA={{180}^{\circ }}$.$\Rightarrow \angle DOA={{180}^{\circ }}-{{180}^{\circ }}+\angle COB$.$\Rightarrow \angle DOA=\angle COB$.We use this result to get the required answers.