ABCD is trapezium in which AB parallel cd. if ad=bc, show that angle A = angle B and angle C= angle D
Answers
GLOSSARY
ll = Parallel
ang = short form of "Angle"
deg = shot form of "Degree"
(Seems like Class 9 NCERT Mathematics Ex. 8.1 , Q 12)
SOLUTION (angles 1,2,3 and 4 marked in the attachment)
Given- AB ll CD, AD =BC
To Prove - 1) Angle A = Angle B
2) Angle C = Angle D
Construction - Extend AB and take a point E on the Extended AB, Such that CE ll AD when CE is joined (Diagram attached)
Proof :
AE ll CD (by constn)
AD ll CE (same reason)
⇒AECD is a llgm (each pair of opp. sides are ll)
This implies, AD = EC (opp. sides of llgm)
But AD = BC (given)
Therefore BC = CE
And angle 3 = angle 4 (angles opp. to eq. sides are equal)
But, Angle 1 + angle 4 = 180 deg (Co-Int. angles) - 1
Angle 2 + angle 3 = 180 deg (Linear pair) - 2
On equating equations 1 and 2 :
L1+L4=L2+L3
L1=L2 (L3 AND L4 cancelled since L3=L4)
or ang A = ang B
ang A +ang C = 180 deg (Co- int. angles) - 3
ang B +ang D = 180 deg (same reason) - 4
On equating equations 3 and 4 :
ang A + ang C = ang B + ang D
ang C = ang D (ang A and ang B cancelled since ang A = Ang B)
Hence Proven
***Hope my explanation was unequivocal and was not ambiguous***
