Question

if two lines intersects then vertically opposite angles are equal prove it

please step by step solution​​

Answer 1

Step-by-step explanation:

According to the problem, we need to prove that if two lines intersect each other, then the vertically opposite angles are equal.

Let us draw the two lines AB and CD intersecting at point O.



We know that the sum of angles lying on a straight line is 180∘180∘.

Let us consider the angles on the line CD. So, we get ∠COB+∠BOD=180∘∠COB+∠BOD=180∘.

⇒∠COB=180∘−∠BOD⇒∠COB=180∘−∠BOD ---(1).

Now, let us consider the angles on the line AB. So, we get ∠COB+∠AOC=180∘∠COB+∠AOC=180∘ ---(2).

Let us substitute equation (2) in equation (1).

So, we get 180∘−∠BOD+∠AOC=180∘180∘−∠BOD+∠AOC=180∘.

⇒∠AOC=180∘−180∘+∠BOD⇒∠AOC=180∘−180∘+∠BOD.

⇒∠AOC=∠BOD⇒∠AOC=∠BOD.

From the figure, we can see that the angles ∠AOC∠AOC and ∠BOD∠BOD are vertically opposite angles.

So, we have proved that if two lines intersect each other, then the vertically opposite angles are equal