Question

Помогите пожалуйста.Найдите углы четырехугольника, если суммы его углов, взятых по три, равны 240, 260 и 280.Периметр параллелограмма равен 24 см.Знайдить стороны параллелограмма, если: а) одна из них на 2 см больше другой. б) одна из них в три раза меньше другой. в) сумма трех его сторон равна 17 см.Заранее спасибо.

Answer 1

Обозначим углы 1,2,3,4, тогда <1+<2+<3=240
<2+<3+<4=260
<3+<4+<1=280, сложим данные равенства
2•(<1+<2+<3+<4)+<3=780, сумма углов выпуклого четырёхугольника равна <1+<2+<3+<4=360, тогда 2•360+<3=780, <3=60.
Из второго уравнения вычтем первое, а из третьего второе,
<4-<1=20,<4=<1+20
<1-<2=20,<2=<1-20
<3=60, тогда
<1+<2+<3+<4=<1+(<1-20)+60+(<1+20)=3<1+60=360, <1=100, тогда <4=120, <2=80
Ответ: 100,80,60,120
а) периметр это сумма длин всех сторон, у параллелограмма они попарно равны, тогда сумма смежных сторон равна 12. Пусть меньшая сторона равна x, тогда вторая x+2, x+x+2=12, x=5, x+2=7
Стороны равны 5,5,7,7
б) x+3x=12, x=3, 3x=9
Стороны равны 3,3,9,9
в) пусть стороны равны а и b, тогда а+а+b=17, 2•(a++b)=24, a=12-b
a+a+b=2•a+b=2•(12-b)+b=24-b=17, b=7,a=12-7=5
Стороны равны 5,5,7,7

Answer 2

Finding the angles of a quadrilateral

To find the angles of a quadrilateral, we can use the fact that the sum of the interior angles of any quadrilateral is always 360 degrees. In this case, we are given that the sums of the angles taken three at a time are 240, 260, and 280 degrees.

Let's denote the angles of the quadrilateral as A, B, C, and D. According to the given information, we have the following equations:

A + B + C = 240 B + C + D = 260 C + D + A = 280 To solve these equations, we can use a system of linear equations. Subtracting the second equation from the first equation, we get:

(A + B + C) - (B + C + D) = 240 - 260 A - D = -20

Similarly, subtracting the third equation from the second equation, we get:

(B + C + D) - (C + D + A) = 260 - 280 B - A = -20

Adding these two equations together, we have:

(A - D) + (B - A) = -20 + (-20) -B = -40 B = 40

Substituting the value of B into the second equation, we get:

40 - A = -20 A = 60

Substituting the values of A and B into the first equation, we get:

60 + 40 + C = 240 C = 140

Finally, substituting the values of B and C into the third equation, we get:

40 + D + 60 = 280 D = 180

Therefore, the angles of the quadrilateral are A = 60 degrees, B = 40 degrees, C = 140 degrees, and D = 180 degrees.

Finding the sides of a parallelogram

To find the sides of a parallelogram, we need more information. In the given problem, we are provided with different conditions for the sides of the parallelogram.

a) If one side is 2 cm longer than the other side: Let's denote the shorter side as x cm. According to the given condition, the longer side is x + 2 cm. Since a parallelogram has opposite sides equal in length, the other two sides will also be x cm and x + 2 cm.

b) If one side is three times smaller than the other side: Let's denote the larger side as x cm. According to the given condition, the smaller side is x / 3 cm. Again, since a parallelogram has opposite sides equal in length, the other two sides will also be x cm and x / 3 cm.

c) If the sum of three sides is 17 cm: Let's denote the three sides as x cm, y cm, and z cm. According to the given condition, we have the equation:

x + y + z = 17 Since a parallelogram has opposite sides equal in length, we can assume that x = z. Therefore, the equation becomes:

x + y + x = 17 2x + y = 17

We have two variables and one equation, so we cannot determine the exact values of x, y, and z without additional information.

In summary, the sides of the parallelogram depend on the specific conditions given for each case.