Date post: | 08-Aug-2015 |
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Prove that a quadrilateral is a parallelogram.
Use coordinate geometry with parallelograms.
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.
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.
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°
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
AB = CD AD = CB AC = AC∆ABC ≅ ∆CDA(By sss rule)BAC = DCA(alt. interior
Angels) DAC = BCA(alt.
interior Angels)AB║CDAD ║CB (By cpct)ABCD is a Parallelogram
C
D
B
A
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.
*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.
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
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