In a parallelogram, opposite angles are equal. How to find the acute angle of a parallelogram
A parallelogram is a quadrilateral whose opposite sides parallel, i.e. they lie on parallel lines (Fig. 1).
Theorem 1. On the properties of the sides and angles of a parallelogram. In a parallelogram, opposite sides are equal, opposite angles are equal, and the sum of the angles adjacent to one side of the parallelogram is 180°.
Proof. In this parallelogram ABCD we draw a diagonal AC and get two triangles ABC and ADC (Fig. 2).
These triangles are equal, since ∠ 1 = ∠ 4, ∠ 2 = ∠ 3 (crosswise angles for parallel lines), and side AC is common. From the equality Δ ABC = Δ ADC it follows that AB = CD, BC = AD, ∠ B = ∠ D. The sum of angles adjacent to one side, for example angles A and D, is equal to 180° as one-sided for parallel lines. The theorem has been proven.
Comment. The equality of opposite sides of a parallelogram means that the segments of parallels cut off by parallel ones are equal.
Corollary 1. If two lines are parallel, then all points on one line are at the same distance from the other line.
Proof. Indeed, let a || b (Fig. 3).
Let us draw perpendiculars BA and CD to straight line a from some two points B and C of line b. Since AB || CD, then figure ABCD is a parallelogram, and therefore AB = CD.
The distance between two parallel lines is the distance from an arbitrary point on one of the lines to the other line.
According to what has been proven, it is equal to the length of the perpendicular drawn from some point of one of the parallel lines to the other line.
Example 1. The perimeter of the parallelogram is 122 cm. One of its sides is 25 cm larger than the other. Find the sides of the parallelogram.
Solution. By Theorem 1, opposite sides of a parallelogram are equal. Let's denote one side of the parallelogram by x and the other by y. Then, by condition $$\left\(\begin(matrix) 2x + 2y = 122 \\x - y = 25 \end(matrix)\right.$$ Solving this system, we obtain x = 43, y = 18. Thus Thus, the sides of the parallelogram are 18, 43, 18 and 43 cm.
Example 2.
Solution. Let Figure 4 meet the conditions of the problem.
Let us denote AB by x and BC by y. According to the condition, the perimeter of the parallelogram is 10 cm, i.e. 2(x + y) = 10, or x + y = 5. The perimeter of triangle ABD is 8 cm. And since AB + AD = x + y = 5 then BD = 8 - 5 = 3. So BD = 3 cm.
Example 3. Find the angles of the parallelogram, knowing that one of them is 50° greater than the other.
Solution. Let Figure 5 meet the conditions of the problem.
Let us denote the degree measure of angle A by x. Then the degree measure of angle D is x + 50°.
Angles BAD and ADC are one-sided interior angles with parallel lines AB and DC and a secant AD. Then the sum of these named angles will be 180°, i.e.
x + x + 50° = 180°, or x = 65°. Thus, ∠ A = ∠ C = 65°, a ∠ B = ∠ D = 115°.
Example 4. The sides of the parallelogram are 4.5 dm and 1.2 dm. A bisector is drawn from the vertex of an acute angle. What parts does it divide into? big side parallelogram?
Solution. Let Figure 6 meet the conditions of the problem.
AE is the bisector of an acute angle of a parallelogram. Therefore, ∠ 1 = ∠ 2.
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Parallelogram, rectangle, rhombus, square (2019)
1. Parallelogram
Compound word "parallelogram"? And behind it lies a very simple figure.
Well, that is, we took two parallel lines:
Crossed by two more:
And inside there is a parallelogram!
What properties does a parallelogram have?
Properties of a parallelogram.
That is, what can you use if the problem is given a parallelogram?
The following theorem answers this question:
Let's draw everything in detail.
What does it mean first point of the theorem? And the fact is that if you HAVE a parallelogram, then you will certainly
The second point means that if there IS a parallelogram, then, again, certainly:
Well, and finally, the third point means that if you HAVE a parallelogram, then be sure to:
Do you see what a wealth of choice there is? What to use in the problem? Try to focus on the question of the task, or just try everything one by one - some “key” will do.
Now let’s ask ourselves another question: how can we recognize a parallelogram “by sight”? What must happen to a quadrilateral so that we have the right to give it the “title” of a parallelogram?
Several signs of a parallelogram answer this question.
Signs of a parallelogram.
Attention! Begin.
Parallelogram.
Please note: if you found at least one sign in your problem, then you definitely have a parallelogram, and you can use all the properties of a parallelogram.
2. Rectangle
I think that it will not be news to you at all that
First question: is a rectangle a parallelogram?
Of course it is! After all, he has - remember, our sign 3?
And from here, of course, it follows that in a rectangle, like in any parallelogram, the diagonals are divided in half by the point of intersection.
But the rectangle also has one distinctive property.
Rectangle property
Why is this property distinctive? Because no other parallelogram has equal diagonals. Let's formulate it more clearly.
Please note: in order to become a rectangle, a quadrilateral must first become a parallelogram, and then demonstrate the equality of the diagonals.
3. Diamond
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.
AVERAGE 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 like 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 therefore.
Do you see how great it is?!
And again simple:
Exactly the same way.
Pay attention: 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, the equality of diagonals is the 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! This is 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, 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 intersection point: .
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.
Problem 1. One of the angles of the parallelogram is 65°. Find the remaining angles of the parallelogram.
∠C =∠A = 65° as opposite angles of a parallelogram.
∠A +∠B = 180° as angles adjacent to one side of a parallelogram.
∠B = 180° - ∠A = 180° - 65° = 115°.
∠D =∠B = 115° as the opposite angles of a parallelogram.
Answer: ∠A =∠C = 65°; ∠B =∠D = 115°.
Task 2. The sum of two angles of a parallelogram is 220°. Find the angles of the parallelogram.
Since a parallelogram has 2 equal acute angles and 2 equal obtuse angles, we are given the sum of two obtuse angles, i.e. ∠B +∠D = 220°. Then ∠B =∠D = 220° :
2 = 110°.
∠A + ∠B = 180° as angles adjacent to one side of a parallelogram, so ∠A = 180° - ∠B = 180° - 110° = 70°. Then ∠C =∠A = 70°.
Answer: ∠A =∠C = 70°; ∠B =∠D = 110°.
Task 3. One of the angles of a parallelogram is 3 times larger than the other. Find the angles of the parallelogram.
Let ∠A =x. Then ∠B = 3x. Knowing that the sum of the angles of a parallelogram adjacent to one of its sides is 180°, we will create an equation.
x = 180 : 4;
We get: ∠A = x = 45°, and ∠B = 3x = 3 ∙ 45° = 135°.
Opposite angles of a parallelogram are equal, therefore,
∠A =∠C = 45°; ∠B =∠D = 135°.
Answer: ∠A =∠C = 45°; ∠B =∠D = 135°.
Task 4. Prove that if a quadrilateral has two parallel and equal sides, then this quadrilateral is a parallelogram.
Proof.
Let's draw the diagonal BD and consider Δ ADB and Δ CBD.
AD = BC by condition. The BD side is common. ∠1 = ∠2 as internal crosswise lying with parallel (by condition) lines AD and BC and secant BD. Therefore, Δ ADB = Δ CBD on two sides and the angle between them (1st sign of equality of triangles). IN equal triangles the corresponding angles are equal, which means ∠3 =∠4. And these angles are internal angles lying crosswise with straight lines AB and CD and secant BD. This implies that the lines AB and CD are parallel. Thus, in this quadrilateral ABCD, the opposite sides are parallel in pairs, therefore, by definition, ABCD is a parallelogram, which is what needed to be proven.
Task 5. The two sides of a parallelogram are in the ratio 2 : 5, and the perimeter is 3.5 m. Find the sides of the parallelogram.
∙ (AB + AD).
Let's denote one part by x. then AB = 2x, AD = 5x meters. Knowing that the perimeter of the parallelogram is 3.5 m, we create the equation:
2 ∙ (2x + 5x) = 3.5;
2 ∙ 7x = 3.5;
x = 3.5 : 14;
One part is 0.25 m. Then AB = 2 ∙ 0.25 = 0.5 m; AD = 5 ∙ 0.25 = 1.25 m.
Examination.
Perimeter of parallelogram P ABCD = 2 ∙ (AB + AD) = 2 ∙ (0,25 + 1,25) = 2 ∙ 1.75 = 3.5 (m).
Since the opposite sides of the parallelogram are equal, then CD = AB = 0.25 m; BC = AD = 1.25 m.
Answer: CD = AB = 0.25 m; BC = AD = 1.25 m.
A parallelogram is a quadrilateral whose opposite sides are parallel in pairs.
This definition is already sufficient, since the remaining properties of the parallelogram follow from it and are proved in the form of theorems.
- The main properties of a parallelogram are:
- a parallelogram is a convex quadrilateral;
- A parallelogram has opposite sides that are equal in pairs;
- In a parallelogram, opposite angles are equal in pairs;
The diagonals of a parallelogram are divided in half by the point of intersection.
Parallelogram - convex quadrilateral Let us first prove the theorem that a parallelogram is a convex quadrilateral
. A polygon is convex if whichever side of it is extended to a straight line, all other sides of the polygon will be on the same side of this straight line.
Let a parallelogram ABCD be given, in which AB is the opposite side for CD, and BC is the opposite side for AD. Then from the definition of a parallelogram it follows that AB || CD, BC || A.D.
Parallel segments have no common points and do not intersect. This means that CD lies on one side of AB. Since segment BC connects point B of segment AB with point C of segment CD, and segment AD connects other points AB and CD, segments BC and AD also lie on the same side of line AB where CD lies. Thus, all three sides - CD, BC, AD - lie on the same side of AB.
Similarly, it is proved that in relation to the other sides of the parallelogram, the other three sides lie on the same side.
Opposite sides and angles are equal One of the properties of a parallelogram is that In a parallelogram, opposite sides and opposite angles are equal in pairs
. For example, if a parallelogram ABCD is given, then it has AB = CD, AD = BC, ∠A = ∠C, ∠B = ∠D. This theorem is proven as follows.
A parallelogram is a quadrilateral. This means it has two diagonals. Since a parallelogram is a convex quadrilateral, any of them divides it into two triangles. In the parallelogram ABCD, consider the triangles ABC and ADC obtained by drawing the diagonal AC.
In these triangles, side AB corresponds to side CD, and side BC corresponds to AD. Therefore, AB = CD and BC = AD.
Angle B corresponds to angle D, i.e. ∠B = ∠D. Angle A of a parallelogram is the sum of two angles - ∠BAC and ∠CAD. Angle C is equal to ∠BCA and ∠ACD. Since pairs of angles are equal to each other, then ∠A = ∠C.
Thus, it is proven that in a parallelogram opposite sides and angles are equal.
Diagonals are divided in half
Since a parallelogram is a convex quadrilateral, it has two diagonals, and they intersect. Let parallelogram ABCD be given, its diagonals AC and BD intersect at point E. Consider the triangles ABE and CDE formed by them.
These triangles have sides AB and CD equal to the opposite sides of a parallelogram. Angle ABE is equal to angle CDE as crosswise lying with parallel lines AB and CD. For the same reason, ∠BAE = ∠DCE. This means ∆ABE = ∆CDE at two angles and the side between them.
You can also notice that angles AEB and CED are vertical and therefore also equal to each other.
Since triangles ABE and CDE are equal to each other, then all their corresponding elements are equal. Side AE of the first triangle corresponds to side CE of the second, which means AE = CE. Similarly BE = DE. Each pair of equal segments constitutes a diagonal of a parallelogram. Thus it is proven that The diagonals of a parallelogram are bisected by their intersection point.
