Question

Two opposite angles of a parallelogram are\[\left( {3x{\text{ }}-{\text{ }}2} \right)^\circ {\text{ }}and{\text{ }}\left( {50{\text{ }}-{\text{ }}x} \right)^\circ \]. Find the measure of each angle of the parallelogram.
$\left( a \right){\text{ 4}}{{\text{0}}^ \circ },{140^ \circ },{\text{4}}{{\text{0}}^ \circ },{140^ \circ }$
$\left( b \right){\text{ 3}}{{\text{7}}^ \circ },{143^ \circ },{37^ \circ },{143^ \circ }$
$\left( c \right){\text{ 3}}{{\text{5}}^ \circ },{145^ \circ },{35^ \circ },{145^ \circ }$
$\left( d \right){\text{ None of these}}$

Answer 1

Hint:
This type of problem becomes very easy if we know the properties of it. Here in this, we will use the properties of a parallelogram. The first one is opposite sides of a parallelogram are equal and the second one is adjacent angles of a parallelogram are supplementary.

Complete step by step solution:
 As we know that
The opposite side of a parallelogram are equal,
So from the question, we can write it as
\[\left( {3x{\text{ }}-{\text{ }}2} \right)^\circ {\text{ = }}\left( {50{\text{ }}-{\text{ }}x} \right)^\circ \]
Now on solving for the value of$x$, we get
$ \Rightarrow 3x + x = 50 + 2$
And on adding and dividing, we get
$ \Rightarrow x = {13^ \circ }$
Therefore the angle will be ${\left( {3x - 2} \right)^ \circ } = \left( {3 \times 13 - 2} \right) = {37^ \circ }$
And similarly, the other angle will be ${\left( {50 - x} \right)^ \circ } = \left( {50 - 13} \right) = {37^ \circ }$
Now, as we know from the properties of a parallelogram that the adjacent angles of a parallelogram are supplementary.
Therefore, we can write mathematically as
$ \Rightarrow x + {37^ \circ } = {180^ \circ }$
And on calculating, we get
$ \Rightarrow x = {143^ \circ }$
So the other angle is also the same.
Hence, four angles are: ${\text{3}}{{\text{7}}^ \circ },{143^ \circ },{37^ \circ },{143^ \circ }$.

Therefore, the option $\left( b \right)$ will be correct.

Additional information:
It is a four-sided figure, in which the condition is that the opposite sides of the figure are parallel. Its opposite sides are parallel. Its opposite angles are equal and its adjacent angles are a sum of 180 degrees and also the diagonals of the figure intersect each other.

Note:
So we can say that a parallelogram is a quadrilateral having four sides. So for solving it, we see that the only thing needed is you have to know or remember the basic properties of a parallelogram and then you can easily solve it. And one important note to keep in mind is that the quadrilateral and parallelogram both possess different properties.