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Math 2
Proving a Quadrilateral is Special
We proved many things about special quadrilaterals based on their definitions. We
now want to see which properties are enough to go backwards. We start with
parallelograms and then branch out to the other quadrilaterals. Remember, to
disprove something, you only need to provide one counterexample. Keep in mind
the properties of the non-parallelogram quadrilaterals that we studied: trapezoids,
isosceles trapezoids, and kites.
Part I: Ways to prove a parallelogram
Determine whether each of the following is a valid way to prove that a quadrilateral
is a parallelogram.
1. Prove or disprove: If a quadrilateral has two pairs of opposite sides that are
parallel, then it is a parallelogram.
2. Prove or disprove: If a quadrilateral has two pairs of opposite sides that are
equal, then it is a parallelogram.
3. Prove or disprove: If a quadrilateral has one pair of opposite sides that is both
parallel and equal, then it is a parallelogram.
4. Prove or disprove: If a quadrilateral has one pair of opposite sides parallel and
the other pair of opposite sides equal, then it is a parallelogram.
5. Prove or disprove: If a quadrilateral has two pairs of opposite angles that are
equal, then it is a parallelogram.
6. Prove or disprove: If a quadrilateral has two adjacent angles supplementary, and
the other two angles are also supplementary, then the quadrilateral is a
parallelogram.
7. Prove or disprove: If a quadrilateral’s diagonals both bisect each other, then the
quadrilateral is a parallelogram.
Part II: Ways to prove other special quadrilaterals
If you wish to disprove these problems, you may either give a counter-example or
draw a figure that has the given information but is not the suggested special
quadrilateral.
8. Prove or disprove: If a quadrilateral has equal length diagonals, then it is a
rectangle.
9. Prove or disprove: If a quadrilateral has equal length diagonals that bisect each
other, then it is a rectangle.
10. Prove or disprove: If a quadrilateral has perpendicular diagonals, then it is a
rhombus.
11. Prove or disprove: If a quadrilateral has perpendicular diagonals and one
diagonal bisects the other, then it is a rhombus.
12. Prove or disprove: If a quadrilateral has perpendicular diagonals and both
diagonals bisect each other, then it is a rhombus.