Geometry
Name______________________
Quadrilaterals: Definitions & The Parallelogram
Date: Oct. 24
Definitions of Special Quadrilaterals
Parallelogram
A parallelogram is a quadrilateral with
________________________________________
 A rectangle is a parallelogram with ________________________________________
 A rhombus is a parallelogram with
________________________________________
 A square is a parallelogram
________________________________________
Trapezoid
A trapezoid is a quadrilateral with
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 An isos. trapezoid is a trapezoid with ________________________________________
Kite
A kite is a quadrilateral with
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Properties of Parallelogram
Diagram
If a quad is a
, then …
U
1)
A
2)
Q
3)
4)
D
Geometry
Before we simply “accept” these properties, we need to be certain these hold true. Let’s first prove the
following property:
If a quad is a parallelogram, then both pairs of opposite sides are congruent.
Diagram
U
A
Given: QUAD is a
Prove: QU  AD
UA  QD
Statements
Q
D
Reasons
Now, we have just proven a property of a parallelogram. Therefore, this is now a theorem, which we can
use, in addition to the definition ofa parallelogram, when taking information away from a quadrilateral
given to be a parallelogram. This means, for example, that should we be given a parallelogram in the
future, we not only can defend a statement that both pairs of opposite sides of the parallelogram are
parallel (by definition), but we now can defend a statement that both pairs of opposite sides are
congruent by citing this theorem.
Geometry
Minimum Conditions to Prove a Quad is a Parallelogram
The previous scenario had us consider what we can deduce if we are given a parallelogram.
Suppose we now want to prove a quadrilateral is a parallelogram.
When we worked with triangles, we proved triangles were congruent by considering the minimal
conditions necessary to do so. In a somewhat related fashion, we should consider the minimal conditions
necessary to prove a quadrilateral is a parallelogram.
T
Consider the following diagram of a quadrilateral.
Below, let’s make conjectures as to the minimum
conditions necessary to prove a quadrilateral is a
parallelogram:
S
U
R
Minimum Conditions
1) If a quad has two pairs of parallel sides, then the quad is a
2)
3)
4)
5)
. (Definition)
Geometry
Let’s consider proving the following minimum condition:
If both pairs of sides of a quad are congruent, then the quad is a parallelogram.
Note: to prove the quad is a parallelogram, one must prove it is a parallelogram, by definition.
Diagram
Given: RS  UT
ST  RU
T
S
Prove: RSTU is a
U
R
Statements
Reasons
Now, we have just proven a minimum condition for proving a quadrilateral is a parallelogram.
Therefore, this is now a theorem, which we can use, in addition to the definition of a parallelogram,
when proving a given quadrilateral is a parallelogram. This means, for example, that should we be trying
to prove a quadrilateral is a parallelogram in the future, we can not only defend a statement that a
quadrilateral is a parallelogram by showing we have both pairs of opposite sides parallel, but we now
can defend a statement that a quadrilateral is a parallelogram by showin we have both pairs of opposite
sides congruent.