You can put this solution on YOUR website! not sure what you mean by consecutive angles between the parallels of a trapezoid.
what i think you mean is that the top and bottom angles on the left side are supplementary to each other and that the top and bottom angles on the right side are supplemental to each other.
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if you label your trapezoid ABCD starting from the top left and going around clockwise, then you want to prove:
angle DAB is supplementary to angle ADC
and you want to prove:
angle ABC is supplementary to angle DCB
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extend AD upwards to point E to form line segment DE (through A).
extend BC upwards to point F to form line segment CF (through B).
it doesn't matter if DE intersects with CF so don't worry whether they cross or not.
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DE and CF are transversals of the parallel lines AB and DC
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we'll work with DE first.
corresponding angles of parallel lines cut by a transveral are congruent so angle ADC is congruent to angle EAB.
angle EAB is supplementary to angle BAD because the sum of their angles is 180 degrees.
angle ADC is also supplementary to angle BAD because the sum of their angles is 180 degrees as well since angle ADC is congruent to angle EAB.
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this constitutes at least the plan of the proof.
you can embellish with all the proper words as see fit.
similar logic applies to the right side angles using the transversal CF.
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