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

Sure, here is a step-by-step proof for the statement "If two lines intersect each other, then vertically opposite angles are equal."

Step 1: Let's assume two lines AB and CD intersect at point O. This forms four angles at point O: ∠AOC, ∠BOC, ∠AOD, and ∠BOD.

Step 2: According to the property of a straight line, the sum of adjacent angles on a straight line is 180 degrees. So, ∠AOC + ∠BOC = 180 degrees and ∠AOD + ∠BOD = 180 degrees.

Step 3: From step 2, we can say that ∠AOC + ∠BOC = ∠AOD + ∠BOD.

Step 4: If we subtract ∠BOC from both sides of the equation in step 3, we get ∠AOC = ∠AOD.

Step 5: Similarly, if we subtract ∠AOC from both sides of the equation in step 3, we get ∠BOC = ∠BOD.

Step 6: Therefore, we have proved that if two lines intersect each other, then vertically opposite angles are equal.