How to find all the angles of a trapezoid. Angles of an isosceles trapezoid. Similarity of formed trapezoid triangles
The trapezoid is geometric figure, a quadrilateral that has two parallel lines. Other two lines cannot be parallel, in which case it would be a parallelogram.
Types of trapezoids
Trapeziums are of three types: rectangular, when the two corners of the trapezoid are 90 degrees each; equilateral, in which the two lateral lines are equal; versatile, where the lateral lines are of different lengths.
Working with trapezoids, you can learn how to calculate their area, height, line size, as well as figure out how to find the corners of a trapezoid.
Rectangular trapezoid
A rectangular trapezoid has two angles of 90 degrees. The other two angles add up to 180 degrees. Therefore, there is a way how to find the corners of a rectangular trapezoid, knowing the size of one of the corners. Let it be, for example, 26 degrees. It is only necessary to subtract the sum of the known angles from the total sum of the angles of the trapezoid - 360 degrees. 360- (90 + 90 + 26) = 154. The desired angle will be 154 degrees. It can be considered simpler: since two angles are straight, then in total they will be 180 degrees, that is, half of 360; the sum of the oblique angles will also be equal to 180, so 180 -26 = 154 can be calculated easier and faster.
Isosceles trapezoid
The isosceles trapezoid has two equal sides that are not grounds. There are formulas that explain how to find the angles of an isosceles trapezoid.
Calculation 1, if the dimensions of the sides of the trapezoid are given
They are designated by the letters A, B and C: A - the dimensions of the sides, B and C - the dimensions of the base, smaller and larger, respectively. The trapezoid must also be called ABCD. For calculations, it is necessary to draw the height H from the angle B. Formed right triangle BHA, where AH and BH are legs, AB is hypotenuse. Now you can calculate the size of the leg AH. To do this, subtract the smaller one from the larger base of the trapezoid, and divide it in half, i.e. (c-b) / 2.
To find the acute angle of a triangle, use the cos function. Cos of the desired angle (β) will be equal to a / ((c-b) / 2). To find out the size of the angle β, you need to use the arcos function. β = arcos 2а / с-b. Because two angles of an equilateral trapezoid are equal, then they will be: angle BAD = angle CDA = arcos 2a / c-b.
Calculation 2. If the dimensions of the bases of the trapezoid are given.
Given the values of the bases of the trapezoid - a and b, you can use the same method as in the previous solution. From the corner b it is necessary to lower the height h. Having the dimensions of the two legs of the newly created triangle, you can use a similar trigonometric function, only in this case it will be tg. To convert an angle and get its value, you need to use the arctg function. Based on the formulas, we get the sizes of the desired angles:
β = arctan 2h / s-b, and the angle α = 180 - arctan 2h / s-b /
Regular versatile trapezoid
There is a way how to find a larger angle of the trapezoid. To do this, you need to know the dimensions of both sharp corners. Knowing them, and knowing that the sum of the angles at any base of the trapezoid is 180 degrees, we conclude that the desired obtuse angle will consist of the difference 180 - the size of the acute angle. You can also find another obtuse angle of the trapezoid.
Angles of an isosceles trapezoid. Hello! This article will focus on solving problems with a trapezoid. This group of tasks is part of the exam, the tasks are simple. We will calculate the angles of the trapezoid, base and height. The solution of a number of problems is reduced to the solution, as they say: where are we without the Pythagorean theorem?
We will work with an isosceles trapezoid. It has equal sides and angles at the bases. There is an article on the trapezium on the blog,.
Let us note a small and important nuance, which we will not describe in detail in the process of solving the tasks themselves. Look, if we have two bases, then the larger base with the heights lowered to it is divided into three segments - one is equal to the smaller base (these are the opposite sides of the rectangle), the other two are equal to each other (these are legs of equal right-angled triangles):
A simple example: given two bases of an isosceles trapezoid 25 and 65. The larger base is divided into segments as follows:
*And further! In tasks, letter designations are not entered. This is done deliberately so as not to overload the solution with algebraic refinements. I agree that this is mathematically illiterate, but the goal is to convey the essence. And you can always designate vertices and other elements yourself and write down a mathematically correct solution.
Consider the tasks:
27439. The bases of an isosceles trapezoid are equal to 51 and 65. The sides are equal to 25. Find the sine of the acute angle of the trapezoid.
In order to find the angle, you need to build the heights. On the sketch, we denote the data in the condition of the value. The lower base is equal to 65, with heights it is divided into segments 7, 51 and 7:
In a right-angled triangle, we know the hypotenuse and leg, we can find the second leg (the height of the trapezoid) and then calculate the sine of the angle.
According to the Pythagorean theorem, the indicated leg is:
In this way:
Answer: 0.96
27440. The bases of an isosceles trapezoid are 43 and 73. The cosine of the acute angle of the trapezoid is 5/7. Find the side.
Plot heights and mark the data in the value condition, the lower base is divided into segments 15, 43 and 15:
27441. The greater base of an isosceles trapezoid is 34. The lateral side is 14. The sine of an acute angle is (2√10) / 7. Find a smaller base.
Let's build the heights. In order to find a smaller base, we need to find what the segment is equal to, which is a leg in a right-angled triangle (marked in blue):
We can calculate the height of the trapezoid, and then find the leg:
By the Pythagorean theorem, we calculate the leg:
Thus, the lesser base is:
27442. The bases of an isosceles trapezoid are 7 and 51. The tangent of an acute angle is 5/11. Find the height of the trapezoid.
Plot the heights and mark the data in the value condition. The lower base is divided into segments:
What to do? We express the tangent of the angle we know at the base in a right-angled triangle:
27443. The lesser base of an isosceles trapezoid is 23. The height of the trapezoid is 39. The tangent of an acute angle is 13/8. Find a larger base.
We build the heights and calculate what the leg is equal to:
Thus, the larger base will be equal to:
27444. The bases of an isosceles trapezoid are 17 and 87. The height of the trapezoid is 14. Find the tangent of the acute angle.
We build the heights and mark the known values on the sketch. The lower base is divided into segments 35, 17, 35:
By definition of tangent:
77152. The bases of the isosceles trapezoid are 6 and 12. The sine of the acute angle of the trapezoid is 0.8. Find the side.
Let's build a sketch, build heights and mark the known values, the larger base is divided into segments 3, 6 and 3:
Let us express the hypotenuse denoted as x in terms of the cosine:
From the main trigonometric identity, we find cosα
In this way:
27818. What is the greater angle of an isosceles trapezoid, if it is known that the difference of opposite angles is equal to 50 0? Give your answer in degrees.
We know from the geometry course that if we have two parallel straight lines and a secant, that the sum of the inner one-sided angles is 180 0. In our case it is
With the condition it is said that the difference between the opposite angles is 50 0, that is
A trapezoid is a flat four square, in which two opposite sides are parallel. They are called foundations trapezium and the other two sides are the sides trapezium.
Instructions
The problem of finding an arbitrary angle in trapezium requires a sufficient amount of additional data. Consider an example in which two base angles are known trapezium... Given the angles & ang-BAD and & ang-CDA, find the angles & ang-ABC and & ang-BCD. The trapezoid has such a property that the sum of the angles at each side is 180 ° -. Then & ang-ABC = 180 ° - & ang-BAD, and & ang-BCD = 180 ° - & ang-CDA.
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In another problem, equality of sides can be specified trapezium and any additional angles. For example, as in the figure, it can be known that the sides AB, BC and CD are equal, and the diagonal makes an angle & ang-CAD = α-with the lower base. square ABC, it is isosceles, since AB = BC. Then & ang-BAC = & ang-BCA. Let's denote it x for brevity, and & ang-ABC - y. The sum of the angles of any tri square a is equal to 180 ° -, from this it follows that 2x + y = 180 ° -, then y = 180 ° - - 2x. At the same time, from the properties trapezium: y + x + α- = 180 ° - and therefore 180 ° - - 2x + x + α- = 180 ° -. Thus, x = α-. We found two corners trapezium: & ang-BAC = 2x = 2α- and & ang-ABC = y = 180 ° - - 2α-. Since AB = CD by condition, the trapezoid is isosceles or isosceles. Means,
Trapezoid Is a quadrilateral with two parallel sides that are bases and two non-parallel sides that are lateral sides.
There are also names such as isosceles or isosceles.
Is a trapezoid whose lateral corners are straight.
Trapezium elements
a, b - base of the trapezoid(a parallel to b),
m, n - lateral sides trapezium,
d 1, d 2 - diagonals trapezium,
h - height trapezium (a segment connecting the bases and at the same time perpendicular to them),
MN - middle line(a segment connecting the midpoints of the sides).
Trapezium area
- Half the sum of bases a, b and height h: S = \ frac (a + b) (2) \ cdot h
- Through middle line MN and height h: S = MN \ cdot h
- Through the diagonals d 1, d 2 and the angle (\ sin \ varphi) between them: S = \ frac (d_ (1) d_ (2) \ sin \ varphi) (2)
Trapezoid properties
The middle line of the trapezoid
middle line is parallel to the bases, equal to their half-sum and divides each segment with the ends located on straight lines that contain the bases (for example, the height of the figure) in half:
MN || a, MN || b, MN = \ frac (a + b) (2)
The sum of the angles of the trapezoid
The sum of the angles of the trapezoid adjacent to each side is 180 ^ (\ circ):
\ alpha + \ beta = 180 ^ (\ circ)
\ gamma + \ delta = 180 ^ (\ circ)
Equal area trapezoid triangles
Equal, that is, having equal areas, are the line segments and the triangles AOB and DOC formed by the lateral sides.
Similarity of formed trapezoid triangles
Similar triangles are AOD and COB, which are formed by their bases and line segments.
\ triangle AOD \ sim \ triangle COB
Similarity coefficient k is found by the formula:
k = \ frac (AD) (BC)
Moreover, the ratio of the areas of these triangles is equal to k ^ (2).
Ratio of lengths of segments and bases
Each segment connecting the bases and passing through the point of intersection of the trapezoid diagonals is divided by this point in the ratio:
\ frac (OX) (OY) = \ frac (BC) (AD)
This will be true for the height with the diagonals themselves.
Trapezoid problems do not seem difficult in a number of shapes that were studied earlier. how special case a rectangular trapezoid is considered. And when looking for its area, it is sometimes more convenient to divide it into two already familiar ones: a rectangle and a triangle. One has only to think a little, and a solution is sure to be found.
Definition of a rectangular trapezoid and its properties
For an arbitrary trapezoid, the bases are parallel, and the sides can have an arbitrary value of the angles to them. If a rectangular trapezoid is considered, then in it one of the sides is always perpendicular to the bases. That is, the two angles in it will be equal to 90 degrees. Moreover, they always belong to adjacent vertices or, in other words, to one side.
Other angles in a rectangular trapezoid are always sharp and obtuse. Moreover, their sum will always be equal to 180 degrees.
Each diagonal forms a right-angled triangle with its smaller lateral side. And the height, which is drawn from the top with an obtuse angle, divides the figure in two. One is a rectangle and the other is a right triangle. By the way, this side is always equal to the height of the trapezoid.
What designations are accepted in the presented formulas?
All the quantities used in different expressions that describe the trapezoid are convenient to immediately stipulate and present in the table:
Formulas that describe the elements of a rectangular trapezoid
The simplest of these links the height and the smaller side:
A few more formulas for this side of a rectangular trapezoid:
c = d * sinα;
c = (a - b) * tan α;
c = √ (d 2 - (a - b) 2).
The first stems from a right-angled triangle. And it says that the leg to the hypotenuse gives the sine of the opposite angle.
In the same triangle, the second leg is equal to the difference between the two bases. Therefore, the statement is true that equates the tangent of the angle to the ratio of the legs.
From the same triangle, you can derive a formula based on knowledge of the Pythagorean theorem. This is the third recorded expression.
You can write formulas for the other side. There are also three of them:
d = (a - b) / cosα;
d = c / sin α;
d = √ (c 2 + (a - b) 2).
The first two are again obtained from the aspect ratio in the same right-angled triangle, and the second is derived from the Pythagorean theorem.
What formula can you use to calculate the area?
The one that is given for an arbitrary trapezoid. You just need to take into account that the height is the side perpendicular to the bases.
S = (a + b) * h / 2.
These values are not always explicitly given. Therefore, to calculate the area of a rectangular trapezoid, you will need to perform some mathematical calculations.
What if you need to calculate the diagonals?
In this case, you need to see that they form two right-angled triangles. Hence, you can always use the Pythagorean theorem. Then the first diagonal will be expressed like this:
d1 = √ (c 2 + b 2)
or in another way, replacing "c" with "h":
d1 = √ (h 2 + b 2).
The formulas for the second diagonal are obtained in a similar way:
d2 = √ (c 2 + b 2) or d 2 = √ (h 2 + a 2).
Problem number 1
Condition... The area of a rectangular trapezoid is known and equal to 120 dm 2. Its height is 8 dm. It is necessary to calculate all sides of the trapezoid. An additional condition is that one base is less than the other by 6 dm.
Decision. Since a rectangular trapezoid is given, in which the height is known, we can immediately say that one of the sides is equal to 8 dm, that is, the smaller side.
Now you can count the other: d = √ (c 2 + (a - b) 2). Moreover, here both side c and the difference of bases are immediately given. The latter is equal to 6 dm, this is known from the condition. Then d will be equal to the square root of (64 + 36), that is, of 100. This is how one more side was found, equal to 10 dm.
The sum of the bases can be found from the formula for the area. It will be equal to twice the area divided by the height. If you count, it turns out 240/8. This means that the sum of the grounds is 30 dm. On the other hand, their difference is 6 dm. By combining these equations, both bases can be calculated:
a + b = 30 and a - b = 6.
You can express a as (b + 6), substitute it in the first equality. Then it turns out that 2b will be equal to 24. Therefore, simply b will turn out to be 12 dm.
Then the last side a is equal to 18 dm.
Answer. Sides of a rectangular trapezoid: a = 18 dm, b = 12 dm, c = 8 dm, d = 10 dm.
Problem number 2
Condition. A rectangular trapezoid is given. Its large flank is equal to the sum of the bases. Its height is 12 cm. A rectangle is constructed, the sides of which are equal to the bases of the trapezoid. It is necessary to calculate the area of this rectangle.
Decision. You need to start with what you are looking for. The required area is determined as the product of a and b. Both of these values are unknown.
You will need to use additional equalities. One of them is based on the statement from the condition: d = a + b. It is necessary to use the third formula for this side, which is given above. It turns out: d 2 = c 2 + (a - b) 2 or (a + b) 2 = c 2 + (a - b) 2.
It is necessary to make transformations by substituting instead of c its value from the condition - 12. After opening the brackets and reducing similar terms, it turns out that 144 = 4 ab.
At the beginning of the solution, it was said that a * b gives the required area. Therefore, in the last expression, you can replace this product with S. A simple calculation will give the area value. S = 36 cm 2.
Answer. The required area is 36 cm 2.
Problem number 3
Condition. The area of a rectangular trapezoid is 150√3 cm². The acute angle is 60 degrees. The angle between the small base and the smaller diagonal has the same meaning. You need to calculate the smaller diagonal.
Decision. From the property of the angles of the trapezoid, it turns out that its obtuse angle is 120º. Then the diagonal divides it into equal parts, because one part of it is already 60 degrees. Then the angle between this diagonal and the second base is also 60 degrees. That is, a triangle formed by a large base, an inclined side and a smaller diagonal is equilateral. Thus, the required diagonal will be equal to a, as well as the lateral side d = a.
Now we need to consider a right-angled triangle. In it, the third angle is 30 degrees. This means that the leg opposite to it is equal to half of the hypotenuse. That is, the smaller base of the trapezoid is equal to half of the desired diagonal: b = a / 2. From it, you need to find a height equal to the lateral side perpendicular to the bases. Side with leg here. From the Pythagorean theorem:
c = (a / 2) * √3.
Now all that remains is to substitute all the values into the area formula:
150√3 = (a + a / 2) * (a / 2 * √3) / 2.
The solution to this equation gives the root 20
Answer. The smaller diagonal is 20 cm long.
