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How to prove that the ABCD parallelogram

How to prove that the ABCD parallelogram

The geometry is completely built on the theorems and proofs.

To prove that an arbitrary figure ABCD is a parallelogram, it is necessary to know the definition and characteristics of this figure.



Parallelogram geometry is called a figure withthe four corners, which are parallel to the opposite side. Thus, a diamond, square and rectangle are the varieties of this quadrangle.


Prove that two of the opposite sides are equaland parallel to each other. In the parallelogram ABCD is the sign looks like this: AB = CD and AB || CD. Draw diagonal AC. These triangles are equal on the basis of a second. AC - a common side, the angles BAC and ACD, as well as the ICA and the product range CAD, are both lying crosswise with parallel lines AB and CD (given in the subject). But as these lying crosswise angles apply to the BC and AD parties, then these segments also lie on parallel lines, that was subjected to the proof.


An important proof of the elements that ABCDparallelogram are diagonal, as in the figure at the intersection of the point O, they are divided into equal segments (AO = OC, BO = OD). The triangles AOB and COD are equal, as are their part in connection with these terms and vertical angles. From this it follows that the angles DBA and CDB well as CAB and ACD equal.


But these angles are lying crosswise in thatthat the lines AB and CD are parallel, and performs the role of the diagonal transversal. Proving such a way that the other two triangles formed by the diagonals are equal, you get that this quadrilateral parallelogram.


Another property for which it can be provedthat the quadrilateral ABCD - parallelogram is: the opposite corners of the figure are equal, that is, the angle B is equal to angle D, and C is equal to the angle A. Sum of angles of triangles, we obtain, if hold the diagonal AC, is equal to 180 °. From this we see that the sum of all the angles of this shape ABCD is 360 °.


Recalling the statements of the problem, it is easy to understand thatA corner angle D and add up to 180 °, angle C is similar to angle D = + 180 °. At the same time, these angles are internal to lie on one side, with their respective straight lines and intersecting. It follows that the lines BC and AD are parallel, and the present figure is a parallelogram.

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