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RECTANGLE:
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A rectangle is a parallelogram with four right angles. |
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Remember: A rectangle has all 6 properties of a parallelogram along with two additional properties: "has four right angles" and "has congruent diagonals."
While the definition contains the word "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.
The properties (theorems) will be stated in "if ...then" form. Both the theorem and its converse (where you swap the "if" and "then" expressions) will be examined.
Click  in the charts below to see each proof.
The * indicates Theorems specifically stated in NY NGMS standards.
This does not imply these are the only theorems that should be studied, but they have been emphasized as important. |
Definition and Theorems pertaining to a rectangle: |
DEFINITION: A rectangle is a parallelogram with four right angles. |
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THEOREM: If a parallelogram is a rectangle, it has congruent diagonals. |
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THEOREM Converse: If a parallelogram has congruent diagonals, it is a rectangle. |
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 In addition, it could be said that :
If a parallelogram has ONE right angle, it is a rectangle.
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Did you know ...
the figure joining, in order, the midpoints of the sides of a rectangle is a rhombus, and the figure joining, in order, the midpoints of the sides of a rhombus is a rectangle.
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RHOMBUS:
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A rhombus is a parallelogram with four congruent sides. |
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Remember: A rhombus has all 6 properties of a parallelogram along with three additional properties: "has four congruent sides", "has diagonals that bisect the angles", and
"has perpendicular diagonals".
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.
The properties (theorems) will be stated in "if ...then" form. Both the theorem and its converse (where you swap the "if" and "then" expressions) will be examined.
Click in the charts below to see each proof.
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Definition and Theorems pertaining to a rhombus: |
DEFINITION: A rhombus is a parallelogram with four congruent sides. |
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THEOREM: If a parallelogram is a rhombus, each diagonal bisects a pair of opposite angles. |
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THEOREM Converse: If a parallelogram has diagonals that bisect a pair of opposite angles, it is a rhombus. |
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| THEOREM: If a parallelogram is a rhombus, the diagonals are perpendicular. |
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THEOREM Converse: If a parallelogram has diagonals that are perpendicular, it is a rhombus. |
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In addition, it could be said that :
If a parallelogram has two adjacent sides congruent, it is a rhombus.
SQUARE:
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A square is a parallelogram with four congruent sides and four right angles. |
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Remember: A square has all 6 properties of a parallelogram along with the two additional properties of a rectangle, and the three additional properties of rhombus.
Since we have already proven properties pertaining to the rectangle and the rhombus, no further proofs will be prepared for the square.
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