Prove that a cyclic trapezium is isosceles and its diagonals are equal

CBSE Class 9 CBSE Class 9 Maths
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1

Prove that a cyclic trapezium is isosceles and its diagonals are equal…

2

let ABCD be the cyclic trapezium with AB IICD.
thru’ C draw CE parallel to AD meeting AB in E.
So
AECD is a parallelogram.
so
angle D=angle AEC… opp angles of a parallelogram are equal…(i)
but
angle D+angle ABC=180… opp angles of a cyclic quadr are supplementary…(ii)
from (i) and (ii)
angle AEC+angle ABC=180
but
angle AEC+angle CEB= 180…linear pair
so
angle ABC= angle CEB …(iii)
so
CE=CB… sides opp equal angles are equal.(iv)
but
CE=AD…opp sides of parallelogram AECD.
from (iv) we get
AD=CB
Thus cyclic quadri ABCD is isoceles.
this proves the first part of the question.
now,
join AC and BD, the diagonals.
in triangles DAB and CBA,
AD=CB…proved before
AB=AB common
angle ADB= angle ACB… angles in the same segment of a circle are equal.here AB is the chord.
so triangles DAB and CBA are congruent…SAS rule.
so
AD=CB… CPCT
hence proved.