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Certified answers contain reliable, trustworthy information vouched for by a hand-picked team of experts. Brainly has millions of high quality answers, all of them carefully moderated by our most trusted community members, but certified answers are the finest of the finest.

Let ABCD be a trapezium where AB//CD and AD=BC

Construction: Draw Perpendicular to AB and CM perpendicular to AB

From triangles ALD and BMC we have

AD=BC(given)

DL=CM(distance between parallel sides)

angle ALD = angle BMC (90 degree)

Therefore by RHS congruence criterion, triangle ALD is congruent to triangle BMC

hence angle DAL = angle CBM (C.PC.T) 1)

Since AB// CD

angle DAL + angle ADC = 180 degree (sum of adjacent interior angles is supplementary)

implies angle CBM + angle ADC = 180 degree(from 1)

Implies ABCD is a cyclic trapezium(SUM of opposite angles is supplementary)

Certified answers contain reliable, trustworthy information vouched for by a hand-picked team of experts. Brainly has millions of high quality answers, all of them carefully moderated by our most trusted community members, but certified answers are the finest of the finest.

Consider a trapezium ABCD with AB *parallel CD and BC =AD *

draw AM perpendicular CD and BM perpendicular to CD

in triangle AMD and BMC

AD = BC given

angle AMD =angle BMC by construction each angle at 90 degree

AM = BM perpendicular distance between two lines is same

triangle AMD congurant to triangle BNC by (RHS rule )

angle ADC = angle BCD by (cpct) ----------1

angle BAD and angle ADC are on same side of transvercal AD

angle ADC + angle BAD = 180 degree ---------2

angle BAD + angle BCD = 180 degree ( using 1st equation )

this equation shows the opposite angles are supplementry

THERE FORE ABCD IS A CYCLIC QUADRILATERALS

HENCE PROVED

draw AM perpendicular CD and BM perpendicular to CD

in triangle AMD and BMC

AD = BC given

angle AMD =angle BMC by construction each angle at 90 degree

AM = BM perpendicular distance between two lines is same

triangle AMD congurant to triangle BNC by (RHS rule )

angle ADC = angle BCD by (cpct) ----------1

angle BAD and angle ADC are on same side of transvercal AD

angle ADC + angle BAD = 180 degree ---------2

angle BAD + angle BCD = 180 degree ( using 1st equation )

this equation shows the opposite angles are supplementry

THERE FORE ABCD IS A CYCLIC QUADRILATERALS

HENCE PROVED