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The converse theorem can be stated as:
A quadrilateral is an isosceles trapezoid if it has two pairs of equal adjacent angles
In quadrilateral ABCD,
< A = < B = a, < C = < D = b
As the sum of all the interior angles of a quadrilateral is 360 degrees, we have
< A + < B + < C + < D = 360
a + a + b + b = 360
2a + 2b = 360
a + b = 180
Thus < A + < D = 180
AB and CD are parallel.
Next we need to prove that AD = BC
Through A draw a line AE parallel to BC.
As quadrilateral ABCE is a parallelogram, AE = BC ........(1)
Since AE parallel to BC, < AED = < C.
Thus triangle ADE is an isosceles triangle
So AD = AE ........(2)
From (1) and (2), we have
AD = BC
So quadrilateral ABCD is an isosceles trapezoid.