Question

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.

Answer 1

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.

Answer 2

Step-by-step explanation:

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