Question 1
If two lines intersect, prove that the vertically opposite angles are equal.
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Solution
Given, two lines AB and CD intersect at point O.
To prove
(i) ∠AOC=∠BOD
(ii) ∠AOD=∠BOC
Proof:
(i) Ray OA stands on line CD. ∴∠AOC+∠AOD=180∘ [linear pair axiom]…(i)
Ray OD stands on line AB. ∴∠AOD+∠BOD=180∘ [linear pair exiom]…(ii)
From the equations (i) and (ii), ⇒∠AOC+∠AOD=∠AOD+∠BOD ∠AOC=∠BOD
(ii) Ray OD stands on line AB. ∴∠AOD+∠BOD=180∘ [linear pair axiom]…(iii)
Ray OB stands on line CD. ∴∠DOB+∠BOC=180∘ …(iv)
From equations (iii) and (iv) ⇒∠AOD+∠BOD=∠DOB+∠BOC ∠AOD=∠BOC
Hence proved.