If diagonals of a trapezium are equal then prove it is cyclic quadrilateral.
Answers
Step-by-step explanation:
How can we prove that trapeziums of equal diagonal are cyclic?
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Girija Warrier
Answered 2 years ago
GIVEN: A trapezium ABCD. AB// CD, diagonals AC = BD. And AC & BD are intersecting at O.
TO PROVE: Trap ABCD is cyclic, ie to prove that any one pair of opposite angles are supplementary. So, let's prove here that<B + <D = 180°
PROOF: AB // CD
=> <DCA= < CAB & < CDB= < DBA ( alternate interior angles formed by // lines.
=> tri OCD ~ tri OAB ( by AAA similarity criterion)
=> OD/OB = OC/OA ( csst)
=> (OD/OB)+1 = (OC/OA)+1
=> (OD+ OB)/OB = (OC+OA)/OA
=> DB/OB= AC/OA
But, DB= AC ( given)
=> OB = OA ………… (1)
=> OC = OD …………(2)
also < AOD = < COB ( Vertically Opposite Angles)) ……………..(3)
By, (1), (2), & (3)
Tri ODA congruent to OCB ( by SAS congruence theorem)
=> < OAD= < OBC = <2
& < ODA = < OCB = < 1 ( both cpct)
Also, < ODC= <OCD = < OAB = < OBA = < 3
Now, by angle sum of quadrilateral
(4 <3) + (2 < 2)+ < (2<1) = 360°
=>{ (2<3) + (<2) +(<1)} = 180°
=> { <B + < D } = 180°
=> trap ABCD is cyclic ( as opposite angles are supplementary)
(Proved)