The rectangle represents a special case of a parallelogram. Any rectangle is a parallelogram, but not each parallelogram – a rectangle. To prove that the parallelogram is a rectangle, it is possible, using signs of equality of triangles.
Instruction
1. Remember definition of a parallelogram. It is a quadrangle which opposite sides are equal and parallel. Besides, the sum of the corners adjacent to one party is equal 180 °. The same property also the rectangle has, only it has to correspond to one more condition. The corners adjacent to one party, at it are equal and make everyone 90 °. That is anyway you will need to prove that at the set figure not only the party are parallel and equal, but all corners are straight lines.
2. Draw ABCD parallelogram. Halve the party of AV and put M. Soyedinite's end it with tops of corners of C and D. You need to prove that corners of an IAU and MBD are equal. Their sum, according to definition of a parallelogram, is 180 °. For a start you should prove equality of triangles of an IAU and MBD that is that pieces of MS and MD are equal among themselves.
3. Make one more construction. Halve side of CD and put N end. Consider attentively of what geometrical figures the initial parallelogram consists now. It is made of two parallelograms of AMND and MBCN. It can be presented also to DMB, an IAU and MBD consisting of triangles. The fact that AMND and MBCN are identical parallelepipeds can be proved, proceeding from properties of a parallelepiped. Pieces of AM and MV are equal, pieces of NC and ND are equal too and they represent halves of the opposite sides of a parallelepiped which by definition are identical. Respectively, the MN line will be equal to the parties of AD and VS and is parallel to them. So, at these identical parallelepipeds of diagonal will be equal, that is the piece of MD is equal to a piece of MC.
4. Compare triangles of an IAU and MBD. Remember signs of equality of triangles. They are three, and in this case it is the most convenient to prove equality on three parties. The parties of MA and MB are identical as the point of M is just on AB midpoint. The parties of AD and BC are equal by definition of a parallelogram. You proved equality of the parties of MD and MC in the previous step. That is triangles are equal, and it means that also all their elements are equal, that is the corner of MAD is equal to MVS corner. But these corners prilezhat to one party, that is the sum is them 180 °. Having halved this number, you receive the size of each corner - 90 °. That is all corners of this parallelogram are straight lines, and it means that it represents a rectangle.