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In this article we will look at all the main properties and characteristics of quadrilaterals.

To begin with, I will arrange all types of quadrilaterals in the form of such a summary diagram:

The diagram is remarkable in that the quadrilaterals in each row have ALL THE PROPERTIES OF THE QUADRIlateralS LOCATED ABOVE THEM. Therefore, you need to remember very little.

Trapezoid is a quadrilateral, two sides of which are parallel and the other two are not parallel. Parallel sides are called trapezoid bases, not parallel - sides.

1 . In the trapeze sum of angles adjacent to a side equal to 180°: A+B=180°, C+D=180°

2 . Bisector of any angle of a trapezoid cuts off at its base a segment equal to the side:

3. Bisectors adjacent corners trapezoids intersect at right angles.

4 .Trapezoid is called isosceles, if its sides are equal:

In an isosceles trapezoid

5. Area of ​​a trapezoid equal to the product of half the sum of the bases and the height:

Parallelogram is a quadrilateral with opposite sides pairwise parallel: In a parallelogram:

• opposite sides and opposite angles are equal
• The diagonals of a parallelogram are bisected by their intersection point:

Accordingly, if a quadrilateral has these properties, then it is a parallelogram.

Area of ​​a parallelogram equal to the product of the base and the height:

or the product of the sides and the sine of the angle between them:

:

Rhombus is a parallelogram in which all sides are equal:

• opposite angles are equal
• diagonals are divided in half by their intersection point
  
• diagonals are mutually perpendicular
  
• The diagonals of a rhombus are the bisectors of the angles

Area of ​​a rhombus equal to half the product of diagonals:

or the product of the square of the side and the sine of the angle between the sides:

Let's consider

They are isosceles because

- general. Means

(on three sides). That's why

And these angles are crosswise for straight lines AB and CD and secant AC. Means,

Similarly it is proved that

This means that this quadrilateral is a parallelogram with equal sides, that is, a rhombus. Q.E.D.

Similar tasks:

1. The area of ​​a rhombus is S. Find the area of ​​a quadrilateral whose vertices are the midpoints of the sides of the rhombus.

2. Two circles with centers at points O1 and O2 intersect at points A and A1, and segments AB and AC are their diameters. Find the angles AA1B and AA1C and prove that points B, A1 and C lie on the same straight line.

3. The medians of a triangle with sides 5 cm, 6 cm and 7 cm intersect at point O. Find the distance from point O to the lines containing the sides of the triangle.

4. Quadrilateral ABCD is inscribed in a circle. It is known that angle ABD=30*, angle ACB=30*, angle BDC=20*. Find the angles of quadrilateral ABCD.

(Research problem.) Compare the sum of the lengths of the medians of a triangle with its perimeter.
1) Draw an arbitrary triangle ABC and draw the median VO.
2) On ray BO, lay down the segment OD = BO and connect point D with points A and C. What is the shape of quadrilateral ABCD?
3) Consider triangle ABD. Compare 2m b with the sum BC + AB (m b is the median of VO).
4) Compose similar inequalities for 2m a and 2m c.
5) Using addition of inequalities, estimate the sum m a + m b + m c.