Question
If two lines intersect, prove that the vertically opposite angles are equal.
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Answer
Let two lines $A B$ and $C D$ intersect at point $O$.
To prove: $\angle A O C=\angle B O D\ ($vertically opposite angles$)$
$\angle A O D=\angle B O C \ ($vertically opposite angles$)$
Proof : $(i)$ Since, ray $OA$ stands on the line $CD$.
$\Rightarrow \angle A O C+\angle A O D-180^{\circ} \ldots (1)\ [$Lincar pair axiom$]$
Also, ray $OD$ stands on the line $AB$.
$\angle A O D+\angle B O D=180^{\circ} \ldots (2)\ [$Linear pair axiom$]$
From equations $(1)$ and $(2),$ we get
$\angle A O C+\angle A O D=\angle A O D+\angle B O D$
$\Rightarrow \angle A O C=\angle B O D$
Hence, proved.
$(ii)$ Since, ray $OD$ stands on the line $A B$.
$\therefore \angle A O D+\angle B O D=180^{\circ} \ldots (3) \ [$Linear pair axiom$]$
Also, ray $O B$ stands on the line $C D$.
$\therefore \angle D O B+\angle B O C=180^{\circ} \ldots (4) \ [$linear pair axiom$]$
From equations $(3)$ and $(4),$ we get
$\angle A O D+\angle B O D=\angle B O D+\angle B O C$
$\Rightarrow \angle A O D=\angle B O C$
Hence, proved.
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