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prove that isoceles trapezium are cyclic quadrilateral?

Consider a trapezium ABCD with AB | |CD and BC = AD.( as it is isoceles is given)

Draw AM ? CD and BN ? CD.

In ?AMD and ?BNC,

AD = BC (Given)

?AMD = ?BNC (By construction, each is 90)

AM = BN (Perpendicular distance between two parallel lines is same)

? ?AMD ? ?BNC (RHS congruence rule)

? ?ADC = ?BCD (CPCT) ... (1)

?BAD and ?ADC are on the same side of transversal AD.

?BAD + ?ADC = 180? ... (2)

?BAD + ?BCD = 180? [Using equation (1)]

This equation shows that the opposite angles are supplementary.

Therefore, ABCD is a cyclic quadrilateral.