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Definition: A rectangle is a parallelogram
with four right angles.
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*Rectangle
I
have all of the properties of the parallelogram PLUS
- 4 right angles
- diagonals congruent |
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Using the definition, the properties of
the rectangle
can be "proven" true and become theorems.
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When dealing
with a rectangle, the definition and theorems are stated as ... |
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1.
A rectangle is a
parallelogram with four right angles.
While the definition states "parallelogram", it is sufficient
to say: "A quadrilateral is a rectangle if and only if it has four
right angles.",
since any quadrilateral with four right angles is a
parallelogram. |
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2.
If a parallelogram has one right angle it is a
rectangle. |
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3. A parallelogram is a rectangle
if and only if its diagonals are congruent. |
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Construction
workers use the fact that the diagonals of a rectangle are
congruent (equal) when attempting to build a "square"
footing for a building, a patio, a fenced area, a table top,
etc. Workers measure the diagonals. When the
diagonals of the project are equal the building line is said
to be square. |
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Definition: A
rhombus is a parallelogram with four congruent sides.
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*Rhombus
I
have all of the properties of the parallelogram PLUS
- 4 congruent sides
- diagonals bisect angles
- diagonals perpendicular |
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Using the definition, the properties of
the rhombus
can be "proven" true and become theorems.
|
When dealing
with a rhombus, the definition and theorems are stated as ... |
|
1.
A rhombus is a
parallelogram with four congruent sides.
While the definition states "parallelogram", it is sufficient
to say: "A quadrilateral is a rhombus if and only if it has four
congruent sides.",
since any quadrilateral with four congruent sides is a
parallelogram. |
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2.
If a parallelogram has two consecutive sides
congruent, it is a rhombus. |
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3.
A parallelogram is a rhombus if
and only if each diagonal bisects a pair of opposite angles. |
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4.
A parallelogram is a rhombus if
and only if the diagonals are perpendicular.
(Proof of theorem appears
further down page.) |
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Definition: A
square is a parallelogram with four congruent sides and
four right angles.
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*Square
Hey,
look at me!
I have all of the properties of the parallelogram AND the
rectangle AND the rhombus.
I have it all! |
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Using the definition, the properties of
the rhombus
can be "proven" true and become theorems.
|
When dealing
with a square, the definition is stated as ... |
|
A square is a
parallelogram with four congruent sides and four right angles.
This definition may also be stated as
A quadrilateral is a square if and only if it is a rhombus
and a rectangle. |
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