If the arms of one angle are respectively parallel to the arms of another angle, show that the two angles are either equ

Solution

Given : Two angles ∠ABC and ∠DEF such that BA || ED and BC || EF.

To prove :∠ABC = ∠DEF or ∠ABC + ∠DEF = 180°.

Proof :The arms of the angles may be parallel in the same sense or in the opposite sense. So, three cases arise.

Case I : When both pairs of arms are parallel in same sense: [Figure (i)]

In this case, BA || ED and BC is the transversal.

∴ ∠ABC = ∠1 [corresponding  ∠s]

Again, BC || EFand DE is the transversal

∴ ∠1 = ∠DEF [corresponding  ∠s]

Hence, ∠ABC = ∠DEF.

Case II : When both pairs of arms are parallel in opposite sense [Figure (ii)]

In this case, BA || ED and BC is the transversal

∴ ∠ABC = ∠1 [corresponding ∠s]

Again, FE || BC and ED is the transversal

∴ ∠DEF = ∠1 [Alternate interior ∠s]

Hence, ∠ABC = ∠DEF.

Case III : When one pair of arms parallel in same sense and other pair parallel in opposite sense [Figure (iii)]

In this case, BA || ED and BC is the transversal

∴ ∠EGB = ∠ABC [Alternate interior ∠s]

Now, BC || EF and DE is the transversal

∴ ∠DEF + ∠EGB = 180° [Consecutive Int. ∠s]

⇒ ∠DEF + ∠ABC = 180° [∠EGB = ∠ABC]

Hence, ∠ABC and DEF are supplementary.