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CBSE Class 8 Answered

parallelogram
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Asked by borahbhairabjyoti60 | 17 Mar, 2019, 07:15: PM
answered-by-expert Expert Answer
 
Given : ABCD is a parallelogram. AP, BP, CR and DR are bisectors of angles A, B, C and D respectively.
To prove : PQRS is a rectangle.
Proof : 
AB || DC                                       ABCD is a parallelogram
angle A + angle D = 180°              AD is transversal 
begin mathsize 16px style 1 half open parentheses angle straight A plus angle straight D close parentheses equals 90 degree
fraction numerator begin display style 1 end style over denominator begin display style 2 end style end fraction angle straight A plus fraction numerator begin display style 1 end style over denominator begin display style 2 end style end fraction angle straight D equals 90 degree
angle DAS plus angle ADS equals 90 degree space space... left parenthesis straight i right parenthesis
In space increment ADS comma
angle DAS plus angle ADS plus angle DSA equals 180 degree
angle DSA equals 90 degree space space space space space space space space space space space space space space space space space space space space space space space from space left parenthesis straight i right parenthesis
lines space AP space and space DR space intersect space
angle PSR equals angle DSA space space space space space space space space space space space space space space space space space space vertically space opposite
angle PSR equals 90 degree
similarly space we space can space prove space that
angle SPQ equals 90 degree comma space angle PQR equals 90 degree space and space angle SRQ equals 90 degree
angle PSR equals angle PQR space and space angle SPQ equals angle SRQ space space space space space space space space space space space space space pair space of space opposite space angles space
PQRS space is space straight a space parallelogram
angle PSR equals angle PQR equals angle SPQ equals angle SRQ equals 90 degree
Hence comma space PQRS space is space straight a space rectangle
end style
 
Answered by Sneha shidid | 18 Mar, 2019, 11:38: AM
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