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.
If the opposite angles of a quadrilateral are equal, then the quadrilateral is a parallelogram. This test is the converse of the property that the opposite sides of a parallelogram are equal. If the opposite sides of a convex quadrilateral are equal, then the quadrilateral is a parallelogram.
Prove this result using congruence in the figure to the right, where the diagonal AC has been joined. Then ABCD is a parallelogram because its opposite sides are equal. It also gives a method of drawing the line parallel to a given line through a given point P.
Then PQ. This test turns out to be very useful, because it uses only one pair of opposite sides. If one pair of opposite sides of a quadrilateral are equal and parallel, then the quadrilateral is a parallelogram.
This test for a parallelogram gives a quick and easy way to construct a parallelogram using a two-sided ruler. Draw a 6 cm interval on each side of the ruler. Joining up the endpoints gives a parallelogram. The test is particularly important in the later theory of vectors.
Then the figure ABQP to the right is a parallelogram. Even a simple vector property like the commutativity of the addition of vectors depends on this construction. The parallelogram ABQP shows, for example, that. This test is the converse of the property that the diagonals of a parallelogram bisect each other. If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram:.
This test gives a very simple construction of a parallelogram. Draw two intersecting lines, then draw two circles with different radii centred on their intersection. Join the points where alternate circles cut the lines. This is a parallelogram because the diagonals bisect each other. It also allows yet another method of completing an angle BAD to a parallelogram, as shown in the following exercise.
Complete this to a construction of the parallelogram ABCD , justifying your answer. This means that a rhombus is just a special case of a parallelogram where all four sides happen to also be congruent. Rectangles, parallelograms, and rhombuses are all special quadrilaterals with certain properties. A parallelogram is a quadrilateral that has:.
But a parallelogram is not always a rectangle. To use the site, please enable JavaScript in your browser and reload the page. Enable contrast version. TutorMe Blog. Andrew Lee June 24, It has four vertices. The interior angles formed at the vertices are such that the adjacent angles add up to make supplementary angle pairs.
A quadrilateral is referred to as a parallelogram if its opposite sides are parallel and congruent. The important differences between a rectangle and a parallelogram are listed in the table given below. Once we know the length of the sides and the length of the diagonals of a given square and a rhombus, we can use the following formulas to calculate their area and perimeter. Example 1: Determine the properties of a rectangle that are not similar to that of a parallelogram. Example 2: Calculate the area of a rectangle with the dimensions 8 units and 5 units.
The property of a rectangle that makes it different from a parallelogram is as follows: All the internal angles of a rectangle are equal to 90 degrees.
Rectangles and parallelograms are different in terms of the diagonals as: The diagonals of a rectangle are equal in measure but it is not so in the case of a parallelogram. Since two right angles add to a straight angle, thus we can say that each pair of co-interior angles are supplementary and thus the opposite sides are parallel. Therefore, it can be concluded that opposite sides are equal and parallel, which marks that the given figure is a parallelogram. Also, its diagonals bisect each other.
Both, rectangle as well as parallelograms have their opposite sides equal and parallel. The main point of difference between the two is that sides of rectangles form 90 degrees while this is not the case with parallelograms. Thus, a rectangle can be called a parallelogram as it fulfills all the requirements of a parallelogram but a parallelogram cannot be called a rectangle.
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