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How does Euclidean geometry describe the properties of parallelograms?

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Three theorems of congruent triangles — Euclidean geometry This diagram depicts three basic theorems that triangles are congruent—that is, they are of equal shape and size—if: (1) two sides and the included angle are equal, designated SAS; (2) two angles and the included side are equal, designated ASA; or (3) all three sides are equal, designated SSS.

In Euclidean geometry, a parallelogram is defined as a four-sided plane figure (a quadrilateral) in which both pairs of opposite sides are parallel and equal in length.

Euclidean principles establish several key properties for these figures:
* Angles: Opposite angles are equal.
* Diagonals: The diagonals of a parallelogram bisect each other.
* Congruence: Bisecting a parallelogram along a diagonal divides the figure into two congruent triangles.

Euclid’s Elements, specifically Book I, provides the foundational proofs regarding the area relations of parallelograms and their relationship to parallel lines. Special types of parallelograms are categorized based on their specific attributes:1
* Rectangle: An equiangular parallelogram where all four angles are equal.
* Rhombus: An equilateral parallelogram where all four sides are equal.
* Square: A parallelogram that is both equiangular and equilateral.