Prove that when two lines intersect, the vertically opposite angles are equal.

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Solution

We have to prove $∠ 1 = ∠ 3 & ∠ 2 = ∠ 4$

$∠ 1 + ∠ 2 = 180^{\circ}$ .......(i) [Linear pair]

$\Rightarrow ∠ 1 = 180^{\circ} - ∠ 2$

Again, $∠ 3 + ∠ 2 = 180^{\circ}$ .......(ii) [Linear pair]

$\Rightarrow ∠ 3 = 180^{\circ} - ∠ 2$

Hence, $∠ 1 = ∠ 3$ [from (i) and (ii)]

Similarly, we can prove that $∠ 2 = ∠ 4$ .

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Prove that when two lines intersect, the vertically opposite angles are equal.

We have to prove ∠1=∠3 & ∠2=∠4 ∠1+∠2=180∘ .......(i) [Linear pair] ⇒∠1=180∘−∠2 Again, ∠3+∠2=180∘ .......(ii) [Linear pair] ⇒∠3=180∘−∠2 Hence, ∠1=∠3 [from (i) and (ii)] Similarly, we can prove that ∠2=∠4.