Question

7. If two lines intersect prove that the vertically opposite angles are equal.

Answer 1

Answer:

Let us consider the angles on the line CD. So, we get ∠COB+∠BOD=180∘. ... From the figure, we can see that the angles ∠AOC and ∠BOD are vertically opposite angles. So, we have proved that if two lines intersect each other, then the vertically opposite angles are equal.

Answer 2

How to solve this type questions ?

We are solving the problem of drawing the two lines which are interesting each other at the fixed point we can use the fact that the sum of the angle lying on the straight line is equal to 180° we know that which is also known as Linear pairs

Now solving This questions

From the figure, we know that,

AB and CD intersect each other at point O.

$\implies$Let the two pairs of vertically opposite angles be, 1st pair – ∠AOC and ∠BOD 2nd pair – ∠AOD and ∠BOC

$\implies$ 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.

$\implies$We know that, If a ray lies on a line then the sum of the adjacent angles is equal to 180°.

$\implies$∠AOC + ∠AOD = 180° (By linear pair axiom) … (i)

$\implies$ Similarly, the ray DO lies on the line AOB. ∠AOD + ∠BOD = 180° (By linear pair axiom) … (ii)

$\implies$From equations (i) and (ii), We have, ∠AOC + ∠AOD = ∠AOD + ∠BOD ∠AOC = ∠BOD – – – – (iii)

$\implies$ Similarly, the ray BO lies on the line COD. ∠DOB + ∠COB = 180° (By linear pair axiom) – – – – (iv)

$\implies$Also, the ray CO lies on the line AOB. ∠COB + ∠AOC = 180° (By linear pair axiom) – – – – (v)

$\implies$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.

Hope this answers your question,

Cheers! :)