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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.

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Solution

Given: In two angles ABCandDEFABDE and BCEF
To prove: ABC+DEF=1800orABC=DEF
Construction: Produce the sides DE and EF of DEF, to meet the sides of ABC at H and G.

Proof: In figure (i) BGEH is a quadrilateral BHE=900 and BGE=900
But sum of angles of a quadrilateral is 3600
HBG+HEG=3600(900+900)
=36001800=1800
ABC and DEF are supplementary In figure (ii) in quadrilateral BGEH,
BHE=900 and HEG=900
HBG+HEG=3600(900+900)
= 36001800=1800 ...(i)
But HEF+HEG=1800 ...(ii)
(Linear pair)
From (i) and (ii)
HEF=HBG
DEF=ABC
Hence ABC and DEF are equal or supplementary


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