When GIVEN a parallelogram, the definition and theorems are stated as ...

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10 Οκτ 2013 (πριν από 3 χρόνια και 10 μήνες)

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When GIVEN a parallelogram, the definition and theorems
are stated as ...


A
parallelogram

is a quadrilateral with both pairs of opposite sides
parallel.


If a quadrilateral is a parallelogram, the
2 pairs of opposite sides are
congruent.




If a quadrilateral is a parallelogram, the
2 pairs of opposite angles are
congruent.


If a quadrilateral is a parallelogram, the
consecutive angles are
supplementary.



If a quadrilateral is a parallelogram, the
diagonals bisect each other.


If a quadrilateral is a parallelogram, the
diagonals form two
congruent triangles.




When trying to PROVE a parallelogram, the definition and
theorems are stated as ...

(many of these theorems are converses of the previous theorems)


A
parallelogram

is a quadrilateral with both pairs of opposite sides
parallel.


If both pairs of opposite sides
of a quadrilateral are congruent, the
quadrilateral is a parallelogram.



If both pairs of opposite angles of a quadrilateral are congruent, the
quadrilateral is a parallelogram.


If

one angle is supplementary to both consecutive angles in a
quadrilateral, the quadrilateral is a parallelogram.



If the diagonals of a quadrilateral bisect each other, the quadrilateral
is a parallelogram.



If ONE PAIR of opposite sides of a quadrilateral are BOTH parallel
and congruent, the quadrilateral is a parallelogram.





**

Be sure to remember this last met
hod, as it may save you time when
working a proof.


Proof of Theorem:

If a quadrilateral is a parallelogram, the
2 pairs of
opposite sides are congruent.

(
Remember:


when attempting to prove a theorem to be true,

you cannot use the theorem as a reason

in your proof.)



STATEMENTS

REASONS

1


1

Given

2

Draw segment from


A

to
C

2

Two points determine exactly one line.

3


3

A
parallelogram

is a quadrilateral with both
pairs of opposite sides parallel.

4


4

If two parallel li
nes are cut by a transversal,
the alternate interior angles are congruent.

5


5

Reflexive property:


A quantity is congruent
to itself.

6


6

ASA:


If two angles and the included side of
one triangle are congruent to the
corresponding parts of another triangle, the
triangles are congruent.

7


7

CPCTC:


Corresponding parts of congruent
triangles are congruent.


Proof of Theorem:

If
ONE PAIR

of opposite sides of a quadrilateral
are
BOTH

par
allel and congruent, the quadrilateral is a parallelogram.



(
Remember:


when attempting to prove a theorem to be true,


you cannot use the theorem as a reason in your proof.)



STATEMENTS

REASONS

1


1

Given

2

Draw segment from


A

to
C

2

Two points determine exactly one line.

3


3

If two parallel lines are cut by a transversal,
th
e alternate interior angles are congruent.

4


4

Reflexive property:


A quantity is congruent
to itself.

5


5

SAS~SAS:


If two sides and the included
angle of one triangle are congruent to the
corresponding parts of another triangle, the
triangles are congruent.

6


6

CPCTC:


Corresponding parts of congruent
triangles are congruent.

7


7

If two lines are cut by a transversal and the
alternate interior angles are congruent, the
lines are parallel.

8


8

A
parallelogram

is a quadrilat
eral with both
pairs of opposite sides parallel.