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Conditions for parallelograms

Congruence & properties of shapes

Japanese school year: Junior high 2

What you learn

Learn the conditions that determine whether a quadrilateral is a parallelogram based on opposite sides, opposite angles, and diagonals. This concept is essential for classifying shapes and conducting advanced geometric proofs. Prior understanding of triangle congruence proofs and parallel line properties is required.

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Key points

This is the fundamental definition used to determine if a quadrilateral is a parallelogram. If both pairs of opposite sides are parallel, the quadrilateral is a parallelogram.

AB∥DC,  AD∥BCAB \parallel DC, \; AD \parallel BC

This condition is used to identify a parallelogram from its side lengths. If both pairs of opposite sides are equal in length, the quadrilateral is a parallelogram.

AB=DC,  AD=BCAB = DC, \; AD = BC

This condition is used to identify a parallelogram from its angles. If both pairs of opposite angles are equal, the quadrilateral is a parallelogram.

∠A=∠C,  ∠B=∠D\angle A = \angle C, \; \angle B = \angle D

This condition is used to confirm a parallelogram using only a single pair of sides. If one pair of opposite sides is both parallel and equal in length, the quadrilateral is a parallelogram.

AB∥DC,  AB=DCAB \parallel DC, \; AB = DC

This condition is used to confirm a parallelogram from its diagonals. If the two diagonals bisect each other at their intersection point OO, the quadrilateral is a parallelogram.

OA=OC,  OB=ODOA = OC, \; OB = OD

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