Why are parallelograms important

Clearly, all the angles in this parallelogram which is actually a rectangle are equal to 90 o. Answer: We have proved that when one angle of a parallelogram is 90 0 , the parallelogram is a rectangle. Show that the quadrilateral is a rhombus. First of all, we note that since the diagonals bisect each other, we can conclude that ABCD is a parallelogram.

So the opposite sides are equal. Answer: We have proved that the quadrilateral in which the diagonals bisect each other at right angles is a rhombus. The diagonals of ABCD bisect each other at right angles. Then find the perimeter of ABCD. From Example 2, if the diagonals of a quadrilateral bisect each other at right angles then it becomes a rhombus. There are two important properties of the diagonals of a parallelogram.

The diagonal of a parallelogram divides the parallelogram into two congruent triangles. And the diagonals of a parallelogram bisect each other. The diagonals of a parallelogram are NOT equal. The opposite sides and opposite angles of a parallelogram are equal. A parallelogram is a quadrilateral with opposite sides equal and parallel. The opposite angle of a parallelogram is also equal.

In short, a parallelogram can be considered as a twisted rectangle. It is more of a rectangle, but the angles at the vertices are not right-angles. The square and a rectangle are the two simple examples of a parallelogram.

Hence the flat surfaces of the furniture such as a table, a cot, a plain sheet of A4 paper can all be counted as examples of a parallelogram. The opposite sides of a rectangle are equal and parallel. In an isosceles trapezoid the diagonals are always congruent. The median of a trapezoid is parallel to the bases and is one-half of the sum of measures of the bases.

If one angle is right, then all angles are right. The diagonals of a parallelogram bisect each other. Each diagonal of a parallelogram separates it into two congruent triangles. Each congruence proof uses the diagonals to divide the quadrilateral into triangles, after which we can apply the methods of congruent triangles developed in the module, Congruence. The material in this module is suitable for Year 8 as further applications of congruence and constructions.

Because of its systematic development, it provides an excellent introduction to proof, converse statements, and sequences of theorems. Considerable guidance in such ideas is normally required in Year 8, which is consolidated by further discussion in later years. Indeed, clarity about these ideas is one of the many reasons for teaching this material at school. Most of the tests that we meet are converses of properties that have already been proven.

For example, the fact that the base angles of an isosceles triangle are equal is a property of isosceles triangles. Now the corresponding test for a triangle to be isosceles is clearly the converse statement:.

Remember that a statement may be true, but its converse false. We proved two important theorems about the angles of a quadrilateral:. To prove the first result, we constructed in each case a diagonal that lies completely inside the quadrilateral. To prove the second result, we produced one side at each vertex of the convex quadrilateral. We begin with parallelograms, because we will be using the results about parallelograms when discussing the other figures.

A parallelogram is a quadrilateral whose opposite sides are parallel. To construct a parallelogram using the definition, we can use the copy-an-angle construction to form parallel lines. For example, suppose that we are given the intervals AB and AD in the diagram below.

See the module, Construction. The three properties of a parallelogram developed below concern first, the interior angles, secondly, the sides, and thirdly the diagonals.

The first property is most easily proven using angle-chasing, but it can also be proven using congruence. The opposite angles of a parallelogram are equal. As an example, this proof has been set out in full, with the congruence test fully developed.

Most of the remaining proofs however, are presented as exercises, with an abbreviated version given as an answer.

The opposite sides of a parallelogram are equal. As a consequence of this property, the intersection of the diagonals is the centre of two concentric circles, one through each pair of opposite vertices. Notice that, in general, a parallelogram does not have a circumcircle through all four vertices.

Besides the definition itself, there are four useful tests for a parallelogram. Our first test is the converse of our first property, that the opposite angles of a quadrilateral are equal. Well, it turns out that at least one of these shapes is very important to those of us who lay out gaging setups or select precision measurement tools. But before we get to the benefits, we have to talk a little bit about the principles involved.

A parallelogram has four straight sides. Each of the two pair of opposing sides is of equal length and is parallel. The unique properties of the parallelogram have been applied extensively in industry to accurately transfer mechanical motion from one place to another.

Perhaps the best known application is the pantograph, a four-sided device used by engravers to reproduce an image outline to a user-definable scale. One type of reed spring consists of two parallel blocks connected by two or more steel strips of equal size and stiffness to form a reed-type flexure linkage.

One of the blocks is attached to a fixed surface.