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Quadrilateral - a polygon consisting of four points (vertices) and four segments(sides) pairwise connecting these points.

Today we will consider geometric figure- quadrilateral. From the name of this figure it already becomes clear that this figure has four corners. But the rest of the characteristics and properties of this figure, we will consider below.

What is a quadrilateral

A quadrilateral is a polygon consisting of four points (vertices) and four segments (sides) connecting these points in pairs. The area of ​​a quadrilateral is half the product of its diagonals and the angle between them.

A quadrilateral is a polygon with four vertices, three of which do not lie on the same line.

Types of quadrilaterals

• A quadrilateral whose opposite sides are pairwise parallel is called a parallelogram.
• A quadrilateral in which two opposite sides are parallel and the other two are not is called a trapezoid.
• A quadrilateral with all right angles is a rectangle.
• A quadrilateral with all sides equal is a rhombus.
• A quadrilateral in which all sides are equal and all angles are right is called a square.

The quadrilateral can be:

self-intersecting

non-convex

convex

Self-intersecting quadrilateral is a quadrilateral in which any of its sides have an intersection point (in blue in the figure).

Non-convex quadrilateral is a quadrilateral in which one of the internal angles is more than 180 degrees (indicated in orange in the figure).

Sum of angles any quadrilateral that is not self-intersecting always equals 360 degrees.

Special types of quadrilaterals

Quadrangles can have additional properties, forming special types of geometric shapes:

• Parallelogram
• Rectangle
• Square
• Trapeze
• Deltoid
• Counterparallelogram

Quadrilateral and circle

A quadrilateral inscribed around a circle (a circle inscribed in a quadrilateral).

The main property of the circumscribed quadrilateral:

A quadrilateral can be circumscribed around a circle if and only if the sums of the lengths of opposite sides are equal.

Quadrilateral inscribed in a circle (circle inscribed around a quadrilateral)

Main property of an inscribed quadrilateral:

A quadrilateral can be inscribed in a circle if and only if the sum of the opposite angles is 180 degrees.

Quadrilateral side length properties

Difference modulus of any two sides of a quadrilateral does not exceed the sum of its other two sides.

|a - b| ≤ c + d

|a - c| ≤ b + d

|a - d| ≤ b + c

|b - c| ≤ a + d

|b - d| ≤ a + b

|c - d| ≤ a + b

Important. The inequality is true for any combination of sides of a quadrilateral. The figure is provided solely for ease of understanding.

In any quadrilateral the sum of the lengths of its three sides is not less than the length of the fourth side.

Important. When solving problems within school curriculum one can use the strict inequality (<). Равенство достигается только в случае, если четырехугольник является "вырожденным", то есть три его точки лежат на одной прямой. То есть эта ситуация не попадает под классическое определение четырехугольника.

Definition. A parallelogram is a quadrilateral whose opposite sides are pairwise parallel.

Property. In a parallelogram, opposite sides are equal and opposite angles are equal.

Property. The diagonals of a parallelogram are bisected by the intersection point.

1 sign of a parallelogram. If two sides of a quadrilateral are equal and parallel, then the quadrilateral is a parallelogram.

2 sign of a parallelogram. If the opposite sides of a quadrilateral are equal in pairs, then the quadrilateral is a parallelogram.

3 sign of a parallelogram. If in a quadrilateral the diagonals intersect and the intersection point is bisected, then this quadrilateral is a parallelogram.

Definition. A trapezoid is a quadrilateral in which two sides are parallel and the other two sides are not parallel. Parallel sides are called grounds.

The trapezoid is called isosceles (isosceles) if its sides are equal. In an isosceles trapezoid, the angles at the bases are equal.

A trapezoid with one right angle is called rectangular.

The segment connecting the midpoints of the sides is called midline of the trapezium. The middle line is parallel to the bases and equal to their half-sum.

Definition. A rectangle is a parallelogram with all right angles.

Property. The diagonals of a rectangle are equal.

Rectangle sign. If the diagonals in a parallelogram are equal, then this parallelogram is a rectangle.

Definition. A rhombus is a parallelogram in which all sides are equal.

Property. The diagonals of a rhombus are mutually perpendicular and bisect its angles.

Definition. A square is a rectangle in which all sides are equal.

A square is a particular kind of rectangle, and also a particular kind of rhombus. Therefore, it has all their properties.

Properties:
1. All corners of the square are right

2. The diagonals of the square are equal, mutually perpendicular, the intersection point is divided in half and the corners of the square are divided in half.

Lesson topic

• Definition of a quadrilateral.

Lesson Objectives

• Educational - repetition, generalization and testing of knowledge on the topic: “Quadrangles”; development of basic skills.
• Developing - to develop students' attention, perseverance, perseverance, logical thinking, mathematical speech.
• Educational - through the lesson to cultivate an attentive attitude towards each other, to instill the ability to listen to comrades, mutual assistance, independence.

Lesson objectives

• To form skills in building a quadrilateral using a scale bar and a drawing triangle.
• Check students' ability to solve problems.

Lesson plan

1. History reference. Non-Euclidean geometry.
2. Quadrilateral.
3. Types of quadrilaterals.

Non-Euclidean geometry

Non-Euclidean geometry, geometry similar to geometry Euclid in that it defines the movement of figures, but differs from Euclidean geometry in that one of its five postulates (second or fifth) is replaced by its negation. The denial of one of the Euclidean postulates (1825) was a significant event in the history of thought, for it served as the first step towards theory of relativity.

Euclid's second postulate states that any line segment can be extended indefinitely. Euclid apparently believed that this postulate also contained the statement that the straight line has infinite length. However in "elliptic" geometry any straight line is finite and, like a circle, is closed.

The fifth postulate states that if a line intersects two given lines in such a way that the two interior angles on one side of it are less than two right angles in sum, then these two lines, if extended indefinitely, will intersect on the side where the sum of these angles is less than the sum two straight lines. But in "hyperbolic" geometry, there may exist a line CB (see Fig.), Perpendicular at point C to a given line r and intersecting another line s at an acute angle at point B, but, nevertheless, the infinite lines r and s will never intersect .

From these revised postulates it followed that the sum of the angles of a triangle, equal to 180° in Euclidean geometry, is greater than 180° in elliptic geometry and less than 180° in hyperbolic geometry.

Quadrilateral