Geometry / Unit 7: Polygons and Quadrilaterals
7.3 Proving a Quadrilateral Is a Parallelogram
Notes
A quadrilateral is a parallelogram if any of these is true:
- Both pairs of opposite sides are parallel (the definition).
- Both pairs of opposite sides are congruent.
- Both pairs of opposite angles are congruent.
- The diagonals bisect each other.
- One pair of opposite sides is both congruent and parallel.
Practice
A quadrilateral has one pair of opposite sides that are congruent and parallel. Is it a parallelogram?
One pair of sides both congruent and parallel is enough.
Quadrilateral $ABCD$ has $\overline{AB} \parallel \overline{CD}$ and $\overline{AD} \cong \overline{BC}$. Must it be a parallelogram?
The parallel pair and the congruent pair are different pairs.
For what value of $x$ is $WXYZ$ a parallelogram if $WX = 3x - 2$, $YZ = x + 8$, $XY = 7$ and $ZW = 7$?
Opposite sides must be congruent: $3x - 2 = x + 8$, so $x = 5$.
Ready for more? Take the Unit 7 test.