In parallelogram ABCD, points M and N are marked on sides BC and AD so that BM

univerkov | September 10, 2021 | education

In parallelogram ABCD, points M and N are marked on sides BC and AD so that BM = DN. Prove that the quadrilateral AMCN is a parallelogram.

Let us prove that triangles ABM and CDN are equal.

In a parallelogram, opposite sides are equal, then AB = CD. Section ВМ = DN by condition. Angle ABM = CDN as opposite angles in a parallelogram. Then the triangle ABM is equal to the triangle CDN on two sides and the angle between them. Then AM = CN.

Since ВС = АD as opposite sides of the parallelogram, and ВМ = DN by condition, then МС = AN.

Since in the quadrangle AMCN the opposite sides are pairwise equal, then such a quadrilateral = parallelogram, which was required to prove.

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