Quadrangles, as a special case of polygons, is a very important topic studied in the school course of geometry. The modern program involves familiarization with this material in the eighth grade. As part of schooling, exclusively convex quadrangles are considered. The rest are studied at the level of higher educational institutions.
The study of quadrangles in different programs for the study of geometry is not the same. The order of introduction of the concept depends on the sequence of presentation of material on polygons.
The procedure for studying quadrangles
In one case, a quadrangle is considered as a special case of a polygon, in the other it is defined as a set of four segments and points located at their intersection. In this case, the conditions of non-belonging of any of these three points to the same line, and the absence of intersections, except at the vertices, must be satisfied.
In most schools, quadrangles are studied in the eighth grade. Having studied first the parallelism of the lines, then the theorem on the sum of the angles of the polygon, we pass to the parallelogram. Having examined its features and proving the theorems associated with them, we pass on to other special cases, receiving answers to the questions: which quadrangle is called a square, rhombus, rectangle, and various types of trapeziums.
Another approach is the study of quadrangles when considering the topic of such figures. Here, quadrangles are also sequentially studied starting with a parallelogram. It is determined which quadrilateral is called a rectangle, a trapezoid. And of course, it is examined in detail what shapes the remaining quadrangles can be.
Four-Angle Shape Classification
What quadrangle is called a square? This can be clarified by examining all the figures that are relevant to this in order. The first thing to get into our attention is an object called a parallelogram. It is formed by four straight paired parallel and intersecting. Separately, cases are determined when this occurs at angles of ninety degrees and those in which all segments formed by such intersections have the same length. In conclusion, we find out which quadrangle is called a trapezoid.
Quadrangles called convex
Let us dwell on the concepts of convex as well as non-convex quadrangles. This difference is of great importance, since only the first of them are studied in the school curriculum.
What quadrangle is called convex? In order to understand this sequentially, draw straight lines through all sides of the figure. If in all cases the entire quadrangle lies in one of two half-planes formed by this line, it is convex. Otherwise, respectively, non-convex.
Plain parallelogram
Now consider the main types of convex quadrangles. Let's start with the parallelogram. We have given a definition of this figure above. In addition to the definition, it is worth noting several properties of this convex polygon.
The sides of the parallelogram opposite each other are equal. Opposite angles are also equal to each other.
The intersection of segments called diagonals forms an angle of ninety degrees. If you sum the squares of their lengths, they will make up the sum of the squares of the faces of the figure. Each such segment forms two identical triangles and four isometric ones.
Any two adjacent corners when added will give one hundred eighty degrees.
When stating the fact that a geometric figure has these properties, it can be argued that it is a parallelogram. Thus, we get the signs of this quadrangle, which determine the belonging of the figure to this class.
The area can be found in two ways. The first will be the search for the product of the sine of the angle and the lengths of the sides adjacent to it. The second way is to determine the result of multiplying the lengths of the height and the face lying opposite it.
Rhombus
What quadrangle is called a rhombus? One in which all of its constituent parties are equal. This geometric figure has all the properties and features of a parallelogram. Another property is the fact that a circle always fits into this figure.
A parallelogram whose adjacent sides are equal is uniquely defined as a rhombus. The area can be calculated as the product of the squared side and the sine of one of the corners.
Rectangle
What quadrangle is called a rectangle? One that has angles of ninety degrees. Since it is also a parallelogram, the properties and attributes of this quadrangle extend to it. The following can also be said about the rectangle:
- The diagonals of this figure have the same length.
- The area is determined by multiplying the sides by each other.
- In the case where the angle of the parallelogram is ninety degrees, it can be argued that this is a rectangle.
Square
The next question from those that we will consider in this publication is which quadrangle is called a square. This is a figure with equal sides and angles of ninety degrees. Based on the above parameters, it has all the same properties that a rectangle and a rhombus have. Accordingly, it also has their signs.
The features of the square include the unique properties of lines connecting its opposite vertices and called diagonals. They have one length and intersect at right angles.
The applied value of the square is difficult to overestimate. Due to its versatility, ease of determining the area and size, this figure is widely used as a reference measure. A number raised to the second power is stably called mathematicians squared. Using square units, they measure the area, carry out integration and general approximations of sizes on the plane. This geometric concept is widely used in architecture and landscape design.
Trapezoid
Next, consider which quadrangle is called a trapezoid. This will be a figure having sides parallel to each other, called bases and non-parallel sides defined by the sides. It is formed by four faces and the same number of angles. When these non-parallel segments are equal, the trapezoid is defined as equilateral. If the figure has an angle of ninety degrees, it will be considered rectangular.
Such a quadrangle, which is called a trapezoid, has another special element. The line that connects the centers of the sides is called the middle. Its length can be determined by finding one second result of adding the lengths of the sides, defined as the base of the figure.
In an isosceles trapezoid as well as in an isosceles triangle, the diagonal lengths and angles between the sides and the bases are equal.
Around such a trapezoid a circle description is always possible.
The circle fits into such a figure, the sum of the lengths of the sides of which is the same as the result of adding its bases.
General conclusions on the topic
In conclusion, we can say that in the course of geometry, the question of which quadrangle is called a square is sufficiently accessible and considered in detail. Despite the fact that in different textbooks we can find some differences in the sequence of presentation of the topics indicated above, they all comprehensively cover the topic of quadrangles.