Question

If the arms of one of the angles are respectively parallel to the arms of another angle show that the two angles are either equal or supplementary

Answer 1

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.

oh ho Gaya ☺

Answer 2

Thank you for asking this question:

here is your answer:

BA || ED and BC is a traversal,

∴ ∠EGB = ∠ABC (these are the alternate angles)

BC || EF and DE is a traversal,

∠DEF + ∠EGB = 180° (these are the co interior angles)

∠DEF + ∠ABC = 180° (∴∠EGB = ∠ABC)

Or ∠ABC and ∠DEF are supplementary.

If there is any confusion please leave a comment below.