In Fig., `p` is a transversal to lines `m and n` and `,` `/_2=120^@` and` /_5=60^0` . Prove that `m |\|n`

Explanation

"GIVEN: two angles `∠ABC` and `∠DEF` such that BA is parallel to ED and BC is parallel to EF.
TO PROVE: `∠ABC = ∠DEF` or `∠ABC +∠DEF= 180@`
PROOF: the arms of the angles may be parallel in the same sense or in opposite sense , therefore, three cases arises:
Case1: when both pairs of arms are parallel in same sense.
In this case: BA is parallel to ED and BC is transversal therefore, ∠ABC= ∠1 [corresponding angles] again , BC is parallel to EF and DE is transversal therefore, ∠1= ∠DEF [corresponding angles] hence, `∠ABC= ∠DEF `
Case2: when both pairs of arms are parallel in opposite sense. In this case: BA is parallel to ED and BC is transversal therefore, `∠ABC= ∠1 `[corresponding angles] again , FE is parallel to BC and ED is transversal therefore, `∠DEF= ∠1 `[alternate interior angles]
hence, ∠ABC= ∠DEF
Case3: when one pair of arms are parallel and other pair parallel in opposite.
In this case:
BA is parallel to ED and BC is transversal therefore, `∠EGB= ∠ABC `[alternate interior angles]
Now, BC is parallel to EF and DE is transversal therefore, `∠DEF +∠EGB = 180@ `[co. interior angles]
⇒`∠DEF+∠ABC = 180° [∠EGB=∠ABC] `
hence, `∠ABC` and `∠DEF `are supplementary."