Quadrilaterals

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A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel. It has four sides, four angles, and four vertices, with the special property that opposite sides never meet.

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Theorem 8.1: A diagonal of a parallelogram divides it into two congruent triangles. This can be proved using the ASA (Angle-Side-Angle) congruence rule with alternate angles formed by parallel lines.

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Theorem 8.2: In a parallelogram, opposite sides are equal. This means if ABCD is a parallelogram, then AB = DC and AD = BC. This follows from the congruent triangles formed by the diagonal.

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Theorem 8.3: If each pair of opposite sides of a quadrilateral is equal, then it is a parallelogram. This is the converse statement - if AB = DC and AD = BC in quadrilateral ABCD, then ABCD is a paral…

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Theorem 8.4: In a parallelogram, opposite angles are equal. If ABCD is a parallelogram, then ∠A = ∠C and ∠B = ∠D. This can be verified by measuring angles in any parallelogram.

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Theorem 8.5: If in a quadrilateral, each pair of opposite angles is equal, then it is a parallelogram. This uses the angle sum property of quadrilaterals and properties of parallel lines with transver…

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Theorem 8.6: The diagonals of a parallelogram bisect each other. If AC and BD are diagonals intersecting at O, then OA = OC and OB = OD. Point O is the midpoint of both diagonals.

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Theorem 8.7: If the diagonals of a quadrilateral bisect each other, then it is a parallelogram. This can be proved by showing that opposite sides become parallel when diagonals bisect each other.