What makes parallelogram

It is a quadrilateral where both pairs of opposite sides are parallel. If we have a parallelogram where all sides are congruent then we have what is called a rhombus. The properties of parallelograms can be applied on rhombi. If we have a quadrilateral where one pair and only one pair of sides are parallel then we have what is called a trapezoid. The parallel sides are called bases while the nonparallel sides are called legs.

If the legs are congruent we have what is called an isosceles trapezoid. In an isosceles trapezoid the diagonals are always congruent. The proof has been set out in full as an example, because the overlapping congruent triangles can be confusing. The diagonals of a rectangle are equal. Let ABCD be a rectangle. Thus we can draw a single circle with centre M through all four vertices. If a parallelogram is known to have one right angle, then repeated use of co-interior angles proves that all its angles are right angles.

We can construct a rectangle with given side lengths by constructing a parallelogram with a right angle on one corner. First drop a perpendicular from a point P to a line.

We have shown above that the diagonals of a rectangle are equal and bisect each other. Conversely, these two properties taken together constitute a test for a quadrilateral to be a rectangle.

A quadrilateral whose diagonals are equal and bisect each other is a rectangle. As a consequence of this result, the endpoints of any two diameters of a circle form a rectangle, because this quadrilateral has equal diagonals that bisect each other. Thus we can construct a rectangle very simply by drawing any two intersecting lines, then drawing any circle centred at the point of intersection.

The quadrilateral formed by joining the four points where the circle cuts the lines is a rectangle because it has equal diagonals that bisect each other. The remaining special quadrilaterals to be treated by the congruence and angle-chasing methods of this module are rhombuses, kites, squares and trapezia.

The sequence of theorems involved in treating all these special quadrilaterals at once becomes quite complicated, so their discussion will be left until the module Rhombuses, Kites, and Trapezia.

Each individual proof, however, is well within Year 8 ability, provided that students have the right experiences. The next step in the development of geometry is a rigorous treatment of similarity. This will allow various results about ratios of lengths to be established, and also make possible the definition of the trigonometric ratios. Similarity is required for the geometry of circles, where another class of special quadrilaterals arises, namely the cyclic quadrilaterals, whose vertices lie on a circle.

Special quadrilaterals and their properties are needed to establish the standard formulas for areas and volumes of figures. Later, these results will be important in developing integration.

Theorems about special quadrilaterals will be widely used in coordinate geometry. Rectangles are so ubiquitous that they go unnoticed in most applications. Rectangles have been useful for as long as there have been buildings, because vertical pillars and horizontal crossbeams are the most obvious way to construct a building of any size, giving a structure in the shape of a rectangular prism, all of whose faces are rectangles.

Parallelograms are not as common in the physical world except as shadows of rectangular objects. Their major role historically has been in the representation of physical concepts by vectors. For example, when two forces are combined, a parallelogram can be drawn to help compute the size and direction of the combined force. When there are three forces, we complete the parallelepiped, which is the three-dimensional analogue of the parallelogram.

Katz, Addison-Wesley, Hence ABCD is a parallelogram, because one pair of opposite sides are equal and parallel. Join AM. Then ABCD is a parallelogram because its diagonals bisect each other. The square on each diagonal is the sum of the squares on any two adjacent sides. Since opposite sides are equal in length, the squares on both diagonals are the same. Hence ABCD is rectangle, because it is a parallelogram with one right angle.

Hence A is a right angle, and similarly, B , C and D are right angles. Both of these facts allow us to prove that the figure is indeed a parallelogram. Consecutive Angles in a Parallelogram are Supplementary. We might find that the information provided will indicate that the diagonals of the quadrilateral bisect each other. If so, then the figure is a parallelogram.