Find all the angles of the parallelogram if the difference between two of them is: 1) 70 °, 2) 110 °, 3) 140 °
Let’s take advantage of the fact that the sum of the angles of the parallelogram adjacent to one side is 180 °.
Let us denote by x and y the angles of the parallelogram adjacent to one side, and x will be a large angle, and y – correspondingly smaller.
Then we can write the following relation:
x + y = 180.
1) By the condition of the problem, x – y = 70.
We solve the system of equations:
x + y = 180;
x – y = 70.
Adding the first equation with the second, we get:
x + y + x – y = 180 + 70;
2x = 250;
x = 250/2;
x = 125 °.
Substituting the found value x = 125 into the equation x + y = 180, we get:
125 + y = 180:
y = 180 – 125;
y = 55 °.
Therefore, in this case, the smaller parallelogram angles are 55 ° and the larger parallelogram angles are 125 °.
2) By the condition of the problem, x – y = 110.
We solve the system of equations:
x + y = 180;
x – y = 110.
Adding the first equation with the second, we get:
x + y + x – y = 180 + 110;
2x = 290;
x = 290/2;
x = 145 °.
Substituting the found value x = 145 into the equation x + y = 180, we get:
145 + y = 180:
y = 180 – 145;
y = 35 °.
Therefore, in this case, the smaller parallelogram angles are 35 ° and the larger parallelogram angles are 145 °.
3) By the condition of the problem, x – y = 140.
We solve the system of equations:
x + y = 180;
x – y = 140.
Adding the first equation with the second, we get:
x + y + x – y = 180 + 140;
2x = 320;
x = 320/2;
x = 160 °.
Substituting the found value x = 160 into the equation x + y = 180, we get:
160 + y = 180:
y = 180 – 160;
y = 20 °.
Therefore, in this case, the smaller parallelogram angles are 20 ° and the larger parallelogram angles are 160 °.