Prove that if the two arms of an angle are perpendicular to the two arms of another angle, then the angles are either equal or supplementary.

Prove that if the two arms of an angle are perpendicular to the two arms of another angle, then the angles are either equal or supplementary.

Solution:

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 opp. 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 opp. 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 opp.

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.