show that a cyclic trapezium is isosceles
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Secondary School Math 8 points
Prove that a cyclic trapezium is always an isosceles trapezium
Ask for details Follow Report by Jk1arnashruthi 02.02.2017
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suhanisharma
suhanisharma Ace
et 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.
4.3
16 votes
THANKS
23
Comments Report
raj3954 Helping Hand
Answer:
ABCD is cyclic
Step-by-step explanation:
As angle A + angle D = 180' (AB||CD)
Angle D = Angle C (because AD=BC)
therefore angle A + angle C = 180'
therefore ABCD is cyclic
4.5
2 votes
THANKS
3
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