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The section highlights essential properties of parallelograms, such as how a diagonal divides it into two congruent triangles, the equality of opposite sides and angles, and the characteristics of diagonals in rectangles, rhombuses, and squares. It also includes concepts related to triangles formed by midpoints.
This section summarizes crucial properties of parallelograms and their relationships with triangles:
Congruent Triangles: A diagonal in a parallelogram divides it into two congruent triangles.
Properties of Opposite Sides: Opposite sides of a parallelogram are equal.
Diagonal Bisect: Diagonals of a parallelogram bisect each other.
In a parallelogram, triangles they form, congruent they are, like a perfect norm.
Imagine two friends (triangles) splitting a cake (the parallelogram) with a shared slice (the diagonal) making them identical!
Remember 'Cosmos': Congruent Sides, Opposite Equal, Midpoints Parallel, Opposite Angles Equal, and Sharing Diagonal.
{'example': 'Show that a diagonal of a parallelogram divides it into two congruent triangles.', 'solution': 'Given parallelogram ABCD, diagonal AC divides into ∆ABC and ∆CDA. Using alternate angle properties and the ASA congruence criterion, we conclude that AC = CA (common) and ∠BCA = ∠DAC, resulting in ∆ABC ≅ ∆CDA.'}
Term: Parallelogram
Definition: A quadrilateral in which both pairs of opposite sides are parallel.
A quadrilateral in which both pairs of opposite sides are parallel.
Term: Congruent Triangles
Definition: Triangles that are exactly equal in shape and size.
Triangles that are exactly equal in shape and size.
Term: Diagonal
Definition: A line segment connecting two non-adjacent vertices of a polygon.
A line segment connecting two non-adjacent vertices of a polygon.