prove that quadrangle is isosceles trapezoid How to prove that quadrangle $ABCD$ is a isosceles trapezoid? 
where $AB$ is parallel to $CD$


You are given two facts:
    1. A qualrilateral ABCD is inscribed in a circle.
    2. AB is parallel to CD

Because of the parallel lines, qualrilateral ABCD is, by definition, a trapezoid.
To prove that trapezoid ABCD is isosceles, you need to show that the non-parallel sides BD and AC have equal lengths.  This can be accomplished as follows.  Erase the lines that go to the circle's center. Draw a single line from point A to point C.  Finish the proof by using these three theorems:
    1. If parallel lines are cut by a transversal, 
       then the alternate interior angles measure the same.

    2. In a circle, the arc subtended by an inscribed angle has
       twice the measure of the inscribed angle.

    3, In a circle, if two minor arcs are equal in measure, 
       then their corresponding chords are equal in length.

