How to find a parallelogram. Parallelogram
And again the question: is a rhombus a parallelogram or not?
With full right - a parallelogram, because it has and (remember our feature 2).
And again, since a rhombus is a parallelogram, then it must have all the properties of a parallelogram. This means that in a rhombus, opposite angles are equal, opposite sides are parallel, and the diagonals bisect at the point of intersection.
Properties of a rhombus
Look at the picture:
As in the case of a rectangle, these properties are distinctive, that is, for each of these properties we can conclude that this is not just a parallelogram, but a rhombus.
Signs of a diamond
And again, pay attention: there must be not just a quadrilateral whose diagonals are perpendicular, but a parallelogram. Make sure:
No, of course, although its diagonals are perpendicular, and the diagonal is the bisector of the angles and. But... diagonals are not divided in half by the point of intersection, therefore - NOT a parallelogram, and therefore NOT a rhombus.
That is, a square is a rectangle and a rhombus at the same time. Let's see what happens.
Is it clear why? - rhombus is the bisector of angle A, which is equal to. This means it divides (and also) into two angles along.
Well, it's quite clear: the diagonals of a rectangle are equal; The diagonals of a rhombus are perpendicular, and in general, a parallelogram of diagonals is divided in half by the point of intersection.
MIDDLE LEVEL
Properties of quadrilaterals. Parallelogram
Properties of a parallelogram
Attention! Words " properties of a parallelogram"mean that if in your task There is parallelogram, then all of the following can be used.
Theorem on the properties of a parallelogram.
In any parallelogram:
Let's understand why this is all true, in other words WE'LL PROVE theorem.
So why is 1) true?
If it is a parallelogram, then:
- lying criss-cross
- lying like crosses.
This means (according to criterion II: and - general.)
Well, that’s it, that’s it! - proved.
But by the way! We also proved 2)!
Why? But (look at the picture), that is, precisely because.
Only 3 left).
To do this, you still have to draw a second diagonal.
And now we see that - according to the II characteristic (angles and the side “between” them).
Properties proven! Let's move on to the signs.
Signs of a parallelogram
Recall that the parallelogram sign answers the question “how do you know?” that a figure is a parallelogram.
In icons it's like this:
Why? It would be nice to understand why - that's enough. But look:
Well, we figured out why sign 1 is true.
Well, it's even easier! Let's draw a diagonal again.
Which means:
AND It's also easy. But...different!
Means, . Wow! But also - internal one-sided with a secant!
Therefore the fact that means that.
And if you look from the other side, then - internal one-sided with a secant! And that's why.
Do you see how great it is?!
And again simple:
Exactly the same, and.
Please note: if you found at least one sign of a parallelogram in your problem, then you have exactly parallelogram and you can use everyone properties of a parallelogram.
For complete clarity, look at the diagram:
Properties of quadrilaterals. Rectangle.
Rectangle properties:
Point 1) is quite obvious - after all, sign 3 () is simply fulfilled
And point 2) - very important. So, let's prove that
This means on two sides (and - general).
Well, since the triangles are equal, then their hypotenuses are also equal.
Proved that!
And imagine, equality of diagonals is a distinctive property of a rectangle among all parallelograms. That is, this statement is true^
Let's understand why?
This means (meaning the angles of a parallelogram). But let us remember once again that it is a parallelogram, and therefore.
Means, . Well, of course, it follows that each of them! After all, they have to give in total!
So they proved that if parallelogram suddenly (!) the diagonals turn out to be equal, then this exactly a rectangle.
But! Pay attention! We're talking about parallelograms! Not just anyone a quadrilateral with equal diagonals is a rectangle, and only parallelogram!
Properties of quadrilaterals. Rhombus
And again the question: is a rhombus a parallelogram or not?
With full right - a parallelogram, because it has (Remember our feature 2).
And again, since a rhombus is a parallelogram, it must have all the properties of a parallelogram. This means that in a rhombus, opposite angles are equal, opposite sides are parallel, and the diagonals bisect at the point of intersection.
But there are also special properties. Let's formulate it.
Properties of a rhombus
Why? Well, since a rhombus is a parallelogram, then its diagonals are divided in half.
Why? Yes, that's why!
In other words, the diagonals turned out to be bisectors of the corners of the rhombus.
As in the case of a rectangle, these properties are distinctive, each of them is also a sign of a rhombus.
Signs of a diamond.
Why is this? And look,
That means both These triangles are isosceles.
To be a rhombus, a quadrilateral must first “become” a parallelogram, and then exhibit feature 1 or feature 2.
Properties of quadrilaterals. Square
That is, a square is a rectangle and a rhombus at the same time. Let's see what happens.
Is it clear why? A square - a rhombus - is the bisector of an angle that is equal to. This means it divides (and also) into two angles along.
Well, it's quite clear: the diagonals of a rectangle are equal; The diagonals of a rhombus are perpendicular, and in general, a parallelogram of diagonals is divided in half by the point of intersection.
Why? Well, let's just apply the Pythagorean theorem to...
SUMMARY AND BASIC FORMULAS
Properties of a parallelogram:
- Opposite sides are equal: , .
- Opposite angles are equal: , .
- The angles on one side add up to: , .
- The diagonals are divided in half by the point of intersection: .
Rectangle properties:
- The diagonals of the rectangle are equal: .
- A rectangle is a parallelogram (for a rectangle all the properties of a parallelogram are fulfilled).
Properties of a rhombus:
- The diagonals of a rhombus are perpendicular: .
- The diagonals of a rhombus are the bisectors of its angles: ; ; ; .
- A rhombus is a parallelogram (for a rhombus all the properties of a parallelogram are fulfilled).
Properties of a square:
A square is a rhombus and a rectangle at the same time, therefore, for a square all the properties of a rectangle and a rhombus are fulfilled. And also.
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A parallelogram is a quadrilateral whose opposite sides are parallel in pairs. The area of a parallelogram is equal to the product of its base (a) and height (h). You can also find its area through two sides and an angle and through diagonals.
Properties of a parallelogram
1. Opposite sides are identical
First of all, let's draw the diagonal \(AC\) . We get two triangles: \(ABC\) and \(ADC\).
Since \(ABCD\) is a parallelogram, the following is true:
\(AD || BC \Rightarrow \angle 1 = \angle 2\) like lying crosswise.
\(AB || CD \Rightarrow \angle3 = \angle 4\) like lying crosswise.
Therefore, (according to the second criterion: and \(AC\) is common).
And that means \(\triangle ABC = \triangle ADC\), then \(AB = CD\) and \(AD = BC\) .
2. Opposite angles are identical
According to the proof properties 1 we know that \(\angle 1 = \angle 2, \angle 3 = \angle 4\). Thus the sum of opposite angles is: \(\angle 1 + \angle 3 = \angle 2 + \angle 4\). Considering that \(\triangle ABC = \triangle ADC\) we get \(\angle A = \angle C \) , \(\angle B = \angle D \) .
3. Diagonals are divided in half by the intersection point
By property 1 we know that opposite sides are identical: \(AB = CD\) . Once again, note the crosswise lying equal angles.
Thus it is clear that \(\triangle AOB = \triangle COD\) according to the second sign of equality of triangles (two angles and the side between them). That is, \(BO = OD\) (opposite the angles \(\angle 2\) and \(\angle 1\) ) and \(AO = OC\) (opposite the angles \(\angle 3\) and \( \angle 4\) respectively).
Signs of a parallelogram
If only one feature is present in your problem, then the figure is a parallelogram and you can use all the properties of this figure.
For better memorization, note that the parallelogram sign will answer the following question - “how to find out?”. That is, how to find out that a given figure is a parallelogram.
1. A parallelogram is a quadrilateral whose two sides are equal and parallel
\(AB = CD\) ; \(AB || CD \Rightarrow ABCD\)- parallelogram.
Let's take a closer look. Why \(AD || BC \) ?
\(\triangle ABC = \triangle ADC\) By property 1: \(AB = CD \) , \(\angle 1 = \angle 2 \) lying crosswise when \(AB \) and \(CD \) and the secant \(AC \) are parallel.
But if \(\triangle ABC = \triangle ADC\), then \(\angle 3 = \angle 4 \) (lie opposite \(AD || BC \) (\(\angle 3 \) and \(\angle 4 \) - those lying crosswise are also equal).
The first sign is correct.
2. A parallelogram is a quadrilateral whose opposite sides are equal
\(AB = CD \) , \(AD = BC \Rightarrow ABCD \) is a parallelogram.
Let's consider this sign. Let's draw the diagonal \(AC\) again.
By property 1\(\triangle ABC = \triangle ACD\).
It follows from this that: \(\angle 1 = \angle 2 \Rightarrow AD || BC \) And \(\angle 3 = \angle 4 \Rightarrow AB || CD \), that is, \(ABCD\) is a parallelogram.
The second sign is correct.
3. A parallelogram is a quadrilateral whose opposite angles are equal
\(\angle A = \angle C\) , \(\angle B = \angle D \Rightarrow ABCD\)- parallelogram.
\(2 \alpha + 2 \beta = 360^(\circ) \)(since \(\angle A = \angle C\) , \(\angle B = \angle D\) by condition).
It turns out, . But \(\alpha \) and \(\beta \) are internal one-sided at the secant \(AB \) .
And what \(\alpha + \beta = 180^(\circ) \) also says that \(AD || BC \) .
Lesson topic
- Properties of the diagonals of a parallelogram.
Lesson Objectives
- Get acquainted with new definitions and remember some already studied.
- State and prove the property of the diagonals of a parallelogram.
- Learn to apply the properties of shapes when solving problems.
- Developmental – to develop students’ attention, perseverance, perseverance, logical thinking, mathematical speech.
- Educational - through the lesson, cultivate an attentive attitude towards each other, instill the ability to listen to comrades, mutual assistance, and independence.
Lesson Objectives
- Test students' problem-solving skills.
Lesson Plan
- Opening remarks.
- Repetition of previously studied material.
- Parallelogram, its properties and features.
- Examples of tasks.
- Self-check.
Introduction
"Large scientific discovery provides a solution to a major problem, but in the solution of any problem there is a grain of discovery.”
Property of opposite sides of a parallelogram
A parallelogram has opposite sides that are equal.
Proof.
Let ABCD be the given parallelogram. And let its diagonals intersect at point O.
Since Δ AOB = Δ COD by the first criterion of equality of triangles (∠ AOB = ∠ COD, as vertical ones, AO=OC, DO=OB, by the property of the diagonals of a parallelogram), then AB=CD. In the same way, from the equality of triangles BOC and DOA, it follows that BC = DA. The theorem has been proven.
Property of opposite angles of a parallelogram
In a parallelogram, opposite angles are equal.
Proof.
Let ABCD be the given parallelogram. And let its diagonals intersect at point O.
From what was proven in the theorem about the properties of the opposite sides of a parallelogram Δ ABC = Δ CDA on three sides (AB=CD, BC=DA from what was proven, AC – general). From the equality of triangles it follows that ∠ ABC = ∠ CDA.
It is also proved that ∠ DAB = ∠ BCD, which follows from ∠ ABD = ∠ CDB. The theorem has been proven.
Property of the diagonals of a parallelogram
The diagonals of a parallelogram intersect and are bisected at the point of intersection.
Proof.
Let ABCD be the given parallelogram. Let's draw the diagonal AC. Let's mark the middle O on it. On the continuation of the segment DO, we'll put aside the segment OB 1 equal to DO.
By the previous theorem, AB 1 CD is a parallelogram. Therefore, line AB 1 is parallel to DC. But through point A only one line parallel to DC can be drawn. This means that straight AB 1 coincides with straight AB.
It is also proved that BC 1 coincides with BC. This means that point C coincides with C 1. parallelogram ABCD coincides with parallelogram AB 1 CD. Consequently, the diagonals of the parallelogram intersect and are bisected at the point of intersection. The theorem has been proven.
In textbooks for regular schools(for example, in Pogorelov) it is proven like this: the diagonals divide the parallelogram into 4 triangles. Let's consider one pair and find out - they are equal: their bases are opposite sides, the corresponding angles adjacent to it are equal, like vertical angles with parallel lines. That is, the segments of the diagonals are equal in pairs. All.
Is that all?
It was proven above that the intersection point bisects the diagonals - if it exists. The above reasoning does not prove its very existence in any way. That is, part of the theorem “the diagonals of a parallelogram intersect” remains unproven.
The funny thing is that this part is much harder to prove. This follows, by the way, from more overall result: any convex quadrilateral will have diagonals that intersect, but any non-convex quadrilateral will not.
On the equality of triangles along a side and two adjacent angles (the second sign of equality of triangles) and others.
Thales found an important practical application to the theorem on the equality of two triangles along a side and two adjacent angles. A range finder was built in the harbor of Miletus to determine the distance to a ship at sea. It consisted of three driven pegs A, B and C (AB = BC) and a marked straight line SC, perpendicular to CA. When a ship appeared on the SK straight line, we found point D such that points D, .B and E were on the same straight line. As is clear from the drawing, the distance CD on the ground is the desired distance to the ship.
Questions
- Are the diagonals of a square divided in half by the point of intersection?
- Are the diagonals of a parallelogram equal?
- Are the opposite angles of a parallelogram equal?
- State the definition of a parallelogram?
- How many signs of a parallelogram?
- Can a rhombus be a parallelogram?
List of sources used
- Kuznetsov A.V., mathematics teacher (grades 5-9), Kiev
- "Single state exam 2006. Mathematics. Educational and training materials for preparing students / Rosobrnadzor, ISOP - M.: Intellect-Center, 2006"
- Mazur K. I. “Solving the main competition problems in mathematics of the collection edited by M. I. Skanavi”
- L. S. Atanasyan, V. F. Butuzov, S. B. Kadomtsev, E. G. Poznyak, I. I. Yudina “Geometry, 7 – 9: textbook for educational institutions”
We worked on the lesson
Kuznetsov A.V.
Poturnak S.A.
Evgeniy Petrov
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