A rectangle is known as a special case of a parallelogram since it adheres to the basic classification of a parallelogram, but it has features that set it apart. The diagonals are thus equal, and divides the rectangle into right triangles, whereas the diagonals of a parallelogram are not equal and bisect it into two congruent triangles with angles depending on that of the parallelogram.
Difference Between Parallelogram and Rectangle. Difference Between Similar Terms and Objects. MLA 8 Stratuss, Mardi. Name required. Email required. Please note: comment moderation is enabled and may delay your comment. There is no need to resubmit your comment.
Notify me of followup comments via e-mail. Written by : Mardi Stratuss. Introduction to geometry. New York: Wiley, In contrast, there are many categories of special quadrilaterals.
Apart from cyclic quadrilaterals, these special quadrilaterals and their properties have been introduced informally over several years, but without congruence, a rigorous discussion of them was not possible. 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. The opposite sides of a parallelogram are congruent so we will need two pairs of congruent segments:. The same thing goes wrong in this case but it is interesting to consider and provides an opportunity to study some of the special types of parallelograms. More generally, a quadrilateral with 4 congruent sides is a rhombus. Once the side length is fixed, there are many possible rhombuses with the given side length as the angles can be varied as depicted in the pictures below:.
The first rhombus above is a square while the second one has angles of 60 and degrees. Note that a rhombus is determined by one side length and a single angle: the given side length determines all four side lengths and opposite angles are congruent while adjacent angles are supplementary.
Congruence of parallelograms. 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. More classes on this subject Geometry Quadrilaterals: Angles.
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