The document contains several theorems about the properties of parallelograms: if a quadrilateral has both pairs of opposite sides congruent, opposite angles congruent, or an angle supplementary to its consecutive angles, then it is a parallelogram. It also proves that if the diagonals of a quadrilateral bisect each other or one pair of opposite sides are congruent and parallel, then it is a parallelogram. This is applied to prove that the trays of an opening sewing box must always be parallel.

2.  Prove that a quadrilateral is a parallelogram.
 Use coordinate geometry with parallelograms.

3. Theorem 6.6:
If both pairs of
opposite sides of a
quadrilateral are
congruent, then the
quadrilateral is a
parallelogram.
A
D
B
C
ABCD is a parallelogram.

4. Theorem 6.7:
If both pairs of
opposite angles of
a quadrilateral are
congruent, then the
quadrilateral is a
parallelogram.
A
D
B
C
ABCD is a parallelogram.

5. Theorem 6.8:
If an angle of a
quadrilateral is
supplementary
to both of its
consecutive
angles, then the
quadrilateral is a
parallelogram.
A
D
B
C
ABCD is a parallelogram.
x°
(180 – x)° x°

6. Theorem 6.9:
If the diagonals
of a quadrilateral
bisect each
other, then the
quadrilateral is a
parallelogram.
ABCD is a parallelogram.
A
D
B
C

7. AB = CD
AD = CB
AC = AC
∆ABC ≅ ∆CDA(By sss rule)
∠BAC = ∠DCA(alt. interior
Angels)
∠DAC = ∠BCA(alt. interior
Angels)
AB║CD
AD ║CB (By cpct)
ABCD is a  Parallelogram
C
D
B
A

8.  As the sewing box below is opened, the trays are
always parallel to each other. Why? Prove it
2.75 in. 2.75 in.
2 in.
2 in.

9. *Each pair of hinges are opposite
sides of a quadrilateral.
*The 2.75 inch sides of the
quadrilateral are opposite and
congruent.
* The 2 inch sides are also
opposite and congruent. Because
opposite sides of the
quadrilateral are congruent, it is
a parallelogram.
* By the definition of a
parallelogram, opposite sides are
parallel, so the trays of the
sewing box are always parallel.
2.75 in. 2.75 in.
2 in.
2 in.

10.  Theorem
If one pair of opposite
sides of a quadrilateral are congruent and
parallel, then the quadrilateral is a
parallelogram.
 ABCD is a parallelogram.
A
B C
D

11. BC ║DA
∠DAC = ∠BCA
(alt.interior
angles)
AC = AC(reflex property)
BC = DA(given)
∆BAC ≅ ∆DCA(by sss rule)
AB = CD(By cpct)
ABCD is a  Parallelogram
C
D
B
A