The geometry is completely constructed on theorems and proofs.
To prove that an arbitrary figure ABCD is a parallelogram, you need to know the definition and attributes of this figure.
A parallelogram in geometry is a figure withFour corners, in which the opposite sides are parallel. Thus, a rhombus, a square and a rectangle are varieties of this quadrangle.
Prove that two of the opposite sides are equalAnd are parallel with respect to each other. In the parallelogram ABCD, this sign looks like this: AB = CD and AB || CD. Draw the diagonal of the speaker. The resulting triangles will be equal in the second sign. AC is the common side, the angles of the BAC and the ACD, as well as the BCA and CAD, are both lying in a crosswise direction with parallel lines AB and CD (given in the condition). But since these cross-lying corners also refer to the sides AD and BC, these segments also lie on parallel straight lines, which was proved.
Important elements of the proof that ABCDParallelogram, are diagonals, since in this figure, when intersected at O, they are divided into equal segments (AO = OC, BO = OD). The triangles AOB and COD are equal, since their sides are equal in relation to these conditions and vertical angles. It follows that both the corners of DBA and CDB are the same as CAB and ACD.
But these same angles are cross-lying,That the lines AB and CD are parallel, and the secant plays the role of the diagonal. Proving in this way that the other two triangles formed by the diagonals are equal, you will get that the given quadrilateral is a parallelogram.
Another property that can be proved,That the quadrilateral ABCD - the parallelogram sounds like this: the opposite angles of this figure are equal, that is, the angle B is equal to the angle D, and the angle C is equal to A. The sum of the angles of the triangles that we get, if we draw the diagonal AC, is 180 °. Proceeding from this, we get that the sum of all the angles of this figure ABCD is 360 °.
Recalling the conditions of the problem, it is easy to understand thatAngle A and angle D in the sum are 180 °, similarly, the angle C + angle D = 180 °. At the same time, these angles are internal, lie on one side, with the corresponding straight and secant angles. It follows that the lines BC and AD are parallel, and the reduced figure is a parallelogram.