The sum of all angles of a quadrilateral is 360. An inscribed quadrilateral and its properties. Detailed theory

Today we will consider a geometric figure - a 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 the school curriculum, you can use a 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.

inscribed and circumscribed polygons,

§ 106. PROPERTIES OF INSCRIBED AND SURROUNDED QUADRANGLES.

Theorem 1. The sum of the opposite angles of an inscribed quadrilateral is 180°.

Let quadrilateral ABCD be inscribed in a circle with center O (Fig. 412). It is required to prove that / A+ / C = 180° and / B+ / D = 180°.

/ A, as inscribed in circle O, measures 1/2 BCD.
/ C, as inscribed in the same circle, measures 1/2 BAD.

Therefore, the sum of angles A and C is measured by half the sum of the arcs BCD and BAD; in sum, these arcs make up a circle, that is, they have 360 ​​°.
From here / A+ / C = 360°: 2 = 180°.

Similarly, it is proved that / B+ / D = 180°. However, this can also be derived in another way. We know that the sum of the interior angles of a convex quadrilateral is 360°. The sum of angles A and C is 180°, which means that the sum of the other two angles of the quadrilateral also remains 180°.

Theorem 2(reverse). If the sum of two opposite angles in a quadrilateral is 180° , then a circle can be circumscribed about such a quadrilateral.

Let the sum of opposite angles of quadrilateral ABCD be 180°, namely
/ A+ / C = 180° and / B+ / D = 180° (Fig. 412).

Let us prove that a circle can be circumscribed around such a quadrilateral.

Proof. A circle can be drawn through any 3 vertices of this quadrilateral, for example, through points A, B and C. Where will point D be located?

Point D can take only one of the following three positions: be inside the circle, be outside the circle, be on the circumference of the circle.

Assume that the vertex is inside the circle and takes position D "(Fig. 413). Then in the quadrilateral ABCD" we will have:

/ B+ / D" = 2 d.

Continuing the side AD" to the intersection with the circle at the point E and connecting the points E and C, we obtain the inscribed quadrilateral ABCE, in which, according to the direct theorem

/ B+ / E = 2 d.

From these two equalities follows:

/ D" = 2 d - / B;
/ E=2 d - / B;

/ D" = / E,

but this cannot be, because / D", as external to triangle CD"E, must be greater than angle E. Therefore, point D cannot be inside the circle.

It is also proved that the vertex D cannot occupy the position D" outside the circle (Fig. 414).

It remains to recognize that the vertex D must lie on the circumference of the circle, i.e., coincide with the point E, which means that a circle can be circumscribed near the quadrilateral ABCD.

Consequences. 1. A circle can be circumscribed around any rectangle.

2. A circle can be circumscribed around an isosceles trapezoid.

In both cases, the sum of the opposite angles is 180°.

Theorem 3. In the circumscribed quadrilateral, the sums of opposite sides are equal. Let the quadrilateral ABCD be circumscribed about a circle (Fig. 415), that is, its sides AB, BC, CD and DA are tangent to this circle.

It is required to prove that AB + CD = AD + BC. We denote the points of contact by the letters M, N, K, P. Based on the properties of tangents drawn to a circle from one point (§ 75), we have:

AR = AK;
BP = VM;
DN=DK;
CN=CM.

Let us add these equalities term by term. We get:

AR + BP + DN + CN = AK + BM + DK + SM,

i.e., AB + CD = AD + BC, which was to be proved.

Exercises.

1. In an inscribed quadrilateral, two opposite angles are related as 3: 5,
and the other two are related as 4:5. Determine the magnitude of these angles.

2. In the described quadrangle, the sum of two opposite sides is 45 cm. The remaining two sides are related as 0.2: 0.3. Find the length of these sides.

A convex quadrilateral is a figure consisting of four sides connected to each other at the vertices, forming four angles together with the sides, while the quadrangle itself is always in the same plane relative to the straight line on which one of its sides lies. In other words, the entire figure is on one side of any of its sides.

As you can see, the definition is quite easy to remember.

Basic properties and types

Almost all figures known to us, consisting of four corners and sides, can be attributed to convex quadrilaterals. The following can be distinguished:

1. parallelogram;
2. square;
3. rectangle;
4. trapezoid;
5. rhombus.

All these figures are united not only by the fact that they are quadrangular, but also by the fact that they are also convex. Just look at the diagram:

The figure shows a convex trapezoid. Here you can see that the trapezoid is on the same plane or on one side of the segment. If you carry out similar actions, you can find out that in the case of all other sides, the trapezoid is convex.

Is a parallelogram a convex quadrilateral?

Above is an image of a parallelogram. As can be seen from the figure, parallelogram is also convex. If you look at the figure with respect to the lines on which the segments AB, BC, CD and AD lie, it becomes clear that it is always on the same plane from these lines. The main features of a parallelogram are that its sides are pairwise parallel and equal in the same way as opposite angles are equal to each other.

Now, imagine a square or a rectangle. According to their main properties, they are also parallelograms, that is, all their sides are arranged in pairs in parallel. Only in the case of a rectangle, the length of the sides can be different, and the angles are right (equal to 90 degrees), a square is a rectangle in which all sides are equal and the corners are also right, while the lengths of the sides and angles of a parallelogram can be different.

As a result, the sum of all four corners of the quadrilateral must be equal to 360 degrees. The easiest way to determine this is by a rectangle: all four corners of the rectangle are right, that is, equal to 90 degrees. The sum of these 90-degree angles gives 360 degrees, in other words, if you add 90 degrees 4 times, you get the desired result.

Property of the diagonals of a convex quadrilateral

The diagonals of a convex quadrilateral intersect. Indeed, this phenomenon can be observed visually, just look at the figure:

The figure on the left shows a non-convex quadrilateral or quadrilateral. As you wish. As you can see, the diagonals do not intersect, at least not all of them. On the right is a convex quadrilateral. Here the property of diagonals to intersect is already observed. The same property can be considered a sign of the convexity of the quadrilateral.

Other properties and signs of convexity of a quadrilateral

Specifically, according to this term, it is very difficult to name any specific properties and features. It is easier to isolate according to different kinds of quadrilaterals of this type. You can start with a parallelogram. We already know that this is a quadrangular figure, the sides of which are pairwise parallel and equal. At the same time, this also includes the property of the diagonals of a parallelogram to intersect with each other, as well as the sign of the convexity of the figure itself: the parallelogram is always in the same plane and on one side relative to any of its sides.

So, the main features and properties are known:

1. the sum of the angles of a quadrilateral is 360 degrees;
2. the diagonals of the figures intersect at one point.

Rectangle. This figure has all the same properties and features as a parallelogram, but all its angles are equal to 90 degrees. Hence the name, rectangle.

Square, the same parallelogram, but its corners are right, like a rectangle. Because of this, a square is rarely called a rectangle. But the main distinguishing feature of a square, in addition to those already listed above, is that all four of its sides are equal.

The trapezoid is a very interesting figure.. This is also a quadrilateral and also convex. In this article, the trapezoid has already been considered using the example of a drawing. It is clear that she is also convex. The main difference, and, accordingly, a sign of a trapezoid is that its sides can be absolutely not equal to each other in length, as well as its angles in value. In this case, the figure always remains on the same plane with respect to any of the straight lines that connect any two of its vertices along the segments forming the figure.

Rhombus is an equally interesting figure. Partly a rhombus can be considered a square. A sign of a rhombus is the fact that its diagonals not only intersect, but also divide the corners of the rhombus in half, and the diagonals themselves intersect at right angles, that is, they are perpendicular. If the lengths of the sides of the rhombus are equal, then the diagonals are also divided in half at the intersection.

Deltoids or convex rhomboids (rhombuses) may have different side lengths. But at the same time, both the main properties and features of the rhombus itself and the features and properties of convexity are still preserved. That is, we can observe that the diagonals bisect the corners and intersect at right angles.

Today's task was to consider and understand what convex quadrilaterals are, what they are and their main features and properties. Attention! It is worth recalling once again that the sum of the angles of a convex quadrilateral is 360 degrees. The perimeter of figures, for example, is equal to the sum of the lengths of all segments forming the figure. The formulas for calculating the perimeter and area of ​​quadrilaterals will be discussed in the following articles.

"Circumscribed Circle" we have seen that a circle can be circumscribed around any triangle. That is, for any triangle there is such a circle that all three vertices of the triangle "sit" on it. Like this:

Question: Can the same be said about a quadrilateral? Is it true that there will always be a circle on which all four vertices of the quadrilateral will “sit”?

It turns out that this is NOT TRUE! NOT ALWAYS a quadrilateral can be inscribed in a circle. There is a very important condition:

In our drawing:

Look, the angles and lie opposite each other, which means they are opposite. What about the corners then? Do they also seem to be opposites? Is it possible to take corners and instead of corners and?

Yes, you certainly may! The main thing is that the quadrangle has some two opposite angles, the sum of which will be. The remaining two angles then themselves will also add up. Do not trust? Let's make sure. Look:

Let. Do you remember what the sum of all four angles of any quadrilateral is? Of course, . That is - always! . But, → .

Magic straight!

So remember firmly:

If a quadrilateral is inscribed in a circle, then the sum of any two of its opposite angles is

and vice versa:

If a quadrilateral has two opposite angles whose sum is equal, then such a quadrilateral is inscribed.

We will not prove all this here (if you are interested, look into the next levels of theory). But let's see what this wonderful fact leads to, that the sum of opposite angles of an inscribed quadrilateral is equal.

For example, the question comes to mind, is it possible to describe a circle around a parallelogram? Let's try the "poke method" first.

Somehow it doesn't work.

Now apply the knowledge:

suppose that we somehow managed to fit a circle onto a parallelogram. Then it must certainly be:, that is.

And now let's recall the properties of a parallelogram:

Every parallelogram has opposite angles.

We got that

And what about the corners? Well, the same of course.

Inscribed → →

Parallelogram→ →

Amazing, right?

It turned out that if a parallelogram is inscribed in a circle, then all its angles are equal, that is, it is a rectangle!

And at the same time - the center of the circle coincides with the intersection point of the diagonals of this rectangle. This, so to speak, is attached as a bonus.

Well, that means that we found out that a parallelogram inscribed in a circle - rectangle.

Now let's talk about the trapezoid. What happens if a trapezoid is inscribed in a circle? And it turns out it will isosceles trapezium. Why?

Let the trapezoid be inscribed in a circle. Then again, but because of the parallelism of the lines and.

Hence, we have: → → an isosceles trapezoid.

Even easier than with a rectangle, right? But you need to remember firmly - come in handy:

Let's list the most main statements tangent to a quadrilateral inscribed in a circle:

1. A quadrilateral is inscribed in a circle if and only if the sum of its two opposite angles is
2. Parallelogram inscribed in a circle rectangle and the center of the circle coincides with the point of intersection of the diagonals
3. A trapezoid inscribed in a circle is isosceles.

Inscribed quadrilateral. Average level

It is known that for any triangle there is a circumscribed circle (we proved this in the topic “Circumscribed Circle”). What can be said about the quadrilateral? Here it turns out that NOT EVERY quadrilateral can be inscribed in a circle, but there is this theorem:

A quadrilateral is inscribed in a circle if and only if the sum of its opposite angles is.

In our drawing -

Let's try to understand why? In other words, we will now prove this theorem. But before proving, you need to understand how the assertion itself works. Did you notice the words “then and only then” in the statement? Such words mean that harmful mathematicians have pushed two statements into one.

Deciphering:

1. "Then" means: If a quadrilateral is inscribed in a circle, then the sum of any two of its opposite angles is equal.
2. “Only then” means: If a quadrilateral has two opposite angles, the sum of which is equal, then such a quadrilateral can be inscribed in a circle.

Just like Alice: “I think what I say” and “I say what I think”.

Now let's figure out why both 1 and 2 are true?

First 1.

Let the quadrilateral be inscribed in a circle. We mark its center and draw the radii and. What will happen? Do you remember that an inscribed angle is half the corresponding central angle? If you remember - now applicable, and if not so - look at the topic "Circle. Inscribed angle".

Inscribed

Inscribed

But look: .

We get that if - is inscribed, then

Well, it is clear that and also adds up. (should also be considered).

Now the “vice versa”, that is, 2.

Let it turn out that the sum of any two opposite angles of a quadrilateral is equal. Let's say let

We don't yet know if we can describe a circle around it. But we know for sure that we are guaranteed to be able to describe a circle around a triangle. So let's do it.

If the point did not “sit down” on the circle, then it inevitably turned out to be either outside or inside.

Let's consider both cases.

Let the point be outside first. Then the segment intersects the circle at some point. Connect and. The result is an inscribed (!) quadrilateral.

We already know about him that the sum of his opposite angles is equal, that is, but by condition we have.

It turns out that it should be like this.

But this cannot be in any way, since - the outer corner for and means .

And inside? Let's do a similar thing. Let the point inside.

Then the continuation of the segment intersects the circle at a point. Again - an inscribed quadrilateral, and according to the condition it must be satisfied, but - an external angle for and means, that is, again, it cannot be that.

That is, a point cannot be either outside or inside the circle - which means it is on the circle!

Proved the whole theorem!

Now let's see what good consequences this theorem gives.

Corollary 1

A parallelogram inscribed in a circle can only be a rectangle.

Let's understand why that is. Let the parallelogram be inscribed in a circle. Then it should be done.

But from the properties of a parallelogram, we know that.

And the same, of course, for the angles and.

So the rectangle turned out - all the corners are along.

But, in addition, there is another additional pleasant fact: the center of the circle circumscribed about the rectangle coincides with the intersection point of the diagonals.

Let's understand why. I hope you remember very well that the angle based on the diameter is a right angle.

Diameter,

Diameter

and hence the center. That's all.

Consequence 2

A trapezoid inscribed in a circle is isosceles.

Let the trapezoid be inscribed in a circle. Then.

And also.

Have we discussed everything? Not really. In fact, there is another, "secret" way to recognize an inscribed quadrilateral. We will formulate this method not very strictly (but clearly), but we will prove it only in the last level of the theory.

If in a quadrilateral one can observe such a picture as here in the figure (here the angles “looking” at the side of the points and are equal), then such a quadrilateral is an inscribed one.

This is a very important drawing - in problems it is often easier to find equal angles than the sum of angles and.

Despite the complete lack of rigor in our formulation, it is correct, and moreover, it is always accepted by USE examiners. You should write like this:

“- inscribed” - and everything will be fine!

Do not forget this important sign - remember the picture, and perhaps it will catch your eye in time when solving the problem.

Inscribed quadrilateral. Brief description and basic formulas

If a quadrilateral is inscribed in a circle, then the sum of any two of its opposite angles is

and vice versa:

If a quadrilateral has two opposite angles whose sum is equal, then such a quadrilateral is inscribed.

A quadrilateral is inscribed in a circle if and only if the sum of its two opposite angles is equal.

Parallelogram inscribed in a circle- necessarily a rectangle, and the center of the circle coincides with the point of intersection of the diagonals.

A trapezoid inscribed in a circle is isosceles.

The concept of a polygon

Definition 1

polygon called a geometric figure in a plane, which consists of pairwise interconnected segments, neighboring of which do not lie on one straight line.

In this case, the segments are called polygon sides, and their ends are polygon vertices.

Definition 2

An $n$-gon is a polygon with $n$ vertices.

Types of polygons

Definition 3

If a polygon always lies on one side of any line passing through its sides, then the polygon is called convex(Fig. 1).

Figure 1. Convex polygon

Definition 4

If the polygon lies on opposite sides of at least one straight line passing through its sides, then the polygon is called non-convex (Fig. 2).

Figure 2. Non-convex polygon

The sum of the angles of a polygon

We introduce the theorem on the sum of angles of a -gon.

Theorem 1

The sum of the angles of a convex -gon is defined as follows

\[(n-2)\cdot (180)^0\]

Proof.

Let us be given a convex polygon $A_1A_2A_3A_4A_5\dots A_n$. Connect its vertex $A_1$ to all other vertices of the given polygon (Fig. 3).

Figure 3

With such a connection, we get $n-2$ triangles. Summing their angles, we get the sum of the angles of the given -gon. Since the sum of the angles of a triangle is $(180)^0,$ we get that the sum of the angles of a convex -gon is determined by the formula

\[(n-2)\cdot (180)^0\]

The theorem has been proven.

The concept of a quadrilateral

Using the definition of $2$, it is easy to introduce the definition of a quadrilateral.

Definition 5

A quadrilateral is a polygon with $4$ vertices (Fig. 4).

Figure 4. Quadrilateral

For a quadrilateral, the concepts of a convex quadrilateral and a non-convex quadrilateral are similarly defined. Classical examples of convex quadrangles are a square, a rectangle, a trapezoid, a rhombus, a parallelogram (Fig. 5).

Figure 5. Convex quadrilaterals

Theorem 2

The sum of the angles of a convex quadrilateral is $(360)^0$

Proof.

By Theorem $1$, we know that the sum of the angles of a convex -gon is determined by the formula

\[(n-2)\cdot (180)^0\]

Therefore, the sum of the angles of a convex quadrilateral is

\[\left(4-2\right)\cdot (180)^0=(360)^0\]

The theorem has been proven.