2. Tyler has written an incorrect proof to show that quadrilateral ABCD is a
parallelogram. He knows segments AB and DC are congruent. He also knows
angles ABC and ADC are congruent. Find the mistake in his proof.
Segment AC is congruent to itself, so triangle ABC is congruent to triangle ADC by
Side-Angle-Side Triangle Congruence Theorem. Since the triangles are congruent, so
are the corresponding parts, and so angle DAC is congruent to ACB. In
quadrilateral ABCD, AB is congruent to CD and AD is parallel to CB. Since AD is
parallel to CB, alternate interior angles DAC and BCA are congruent. Since
alternate interior angles are congruent, AB must be parallel to CD. Quadrilateral
ABCD must be a parallelogram since both pairs of opposite sides are parallel.
