No, not every trapezoid is an isosceles trapezoid.
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Not normally but quite possible in the form of an isosceles trapezoid.
An isosceles trapezoid
If the two sides which are not parallel have equal lengths, then the trapezoid is called an isosceles trapezoid (standard trapezoid has unequal sides). The base angles in an isosceles trapezoid are equal in measurement, also I know that a trapezoid with two nonparallel sides the same length is called an isosceles trapezoid.
square, rectangle and an isosceles trapezoid.
An isosceles triangle and a isosceles trapezoid.