Lesson HOW TO solve problems on the angles of parallelograms - Examples

How to solve problems on the angles of parallelograms - Examples

In this lesson you will find the solutions of some typical problems on the angles of parallelograms.
To solve this kind of problems use the basic properties of parallelograms:
    - the opposite angles of a parallelogram are congruent
            (see the lesson Opposite angles of a parallelogram under the topic Parallelograms of the section Geometry in this site), and
    - the sum of the consecutive angles of a parallelogram is equal to 180°
            (see the lesson Consecutive angles of a parallelogram under the same topic of the same section in this site).


Problem 1
In a parallelogram, one angle has the angle measure of 73°. Find other angles of the parallelogram.

Solution
Since the sum of the consecutive angles of a parallelogram is equal to 180°, the two other consecutive angles of the parallelogram have the angle measure
180°- 73° = 107°.

Since the opposite angles of a parallelogram are congruent, the angle opposite to the given one of 73° has the angle measure of 73° too.

Answer. The rest of the angles of the parallelogram have the angle measure of 107° (the consecutive angle), 73° (the opposite angle), and 107° (the second consecutive angle).


Problem 2
In a parallelogram, one angle is in 28° greater than the other.
Find the angles of the parallelogram.

Solution
Let x be the angle measure (in degrees) of the first of the two given angles.
Then the angle measure of the second angle is x - 28 in accordance with the problem condition.

Since the given two angles of the parallelogram are not congruent, they are not opposite. Hence, they are consecutive.

Therefore, you can write the equation
x + (x-28) = 180
saying that the sum of the consecutive angles of a parallelogram is equal to 180°.

Simplify and solve this equation:
2x - 28 = 180,
2x = 180 + 28,
2x = 208,
x = 104.

Thus, the first angle has the angle measure of 104°.
Then the two consecutive angles are 180° - 104° = 76°.
You can check that the first angle is in 28° greater than the consecutive, as it is stated by the condition:
104° - 76° = 28°.

Answer. The angles of the parallelogram are 108°, 76°, 108° and 76°.


Problem 3
In a parallelogram, one angle is four times greater than the other.
Find the angles of the parallelogram.

Solution
Let x be the angle measure (in degrees) of the smaller of the two given angles.
Then the angle measure of the greater angle is 4x in accordance with the problem condition.

Since the given two angles of the parallelogram are not congruent, they are not opposite. Hence, they are consecutive.

Therefore, you can write the equation
x + 4x = 180
saying that the sum of the consecutive angles of a parallelogram is equal to 180°.

Simplify and solve this equation:
5x = 180,
x = 36.

Thus, the smaller angle has the angle measure of 36°.
Then the larger angle has the angle measure of 4*36° = 144° in accordance with the problem condition.
You can check that the sum of these two consecutive angles is equal to 180°:
36° + 144° = 180°.

Answer. The angles of the parallelogram are 36°, 144°, 36° and 144°.