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Prove two line segments of a quadrilateral are equal

Question
In the trapezoid ABCD, AB // CD, angle D = angle C, prove that AD = BC
Proof:
trapezoid example solution figure
Draw a line segment passing through point B and parallel to AD and intersects DC at point E.
Since BE // AD, then angle 1 = angle D.
(Theorem: If two parallel lines are cut by a transversal, then the corresponding angles are equal).
Since angle D = angle C (given), and angle D = angle 1, so angle 1 = angle C, so BE = BC.
(Theorem: If two angles of a triangle are equal, then the sides opposite these angles are also equal.)
Since AB // DC (given), and BE // AD, so BE = AD
(Theorem: Two parallels are equal if they are between other two parallels.)
Since BE = AD and BE = BC, so AD = BC.