Question

The opposite angles of a parallelogram $\left( {3x - 2} \right)$ and $\left( {x + 48} \right)$ . Find the measure of each angle at the parallelogram.

Answer 1

Hint: Use the property of parallelogram according to which the opposite angles are congruent. Therefore, equating the expression $\left( {3x - 2} \right)$ and $\left( {x + 48} \right)$ , solve for the only unknown ‘x’. Now use the value of ‘x’ to get the value of these two angles. Then use the property that the sum of the interior angles of a quadrilateral is $360^\circ $ . This will give us the sum of other pairs of opposite angles and thus the remaining two angles.

Complete step-by-step answer:
Here in the given problem, we are given with the expressions $\left( {3x - 2} \right)$ and $\left( {x + 48} \right)$ for a pair of opposite angles of a parallelogram. Using these expressions we need to find the measure of each of the angles of the parallelogram.
The expression $\left( {3x - 2} \right)$ and $\left( {x + 48} \right)$ are the values for the opposite angles of a parallelogram in terms of a variable $'x'$ . So if we find the value of this variable then we can find the value for these two angles.
For this, we need to use some properties of a parallelogram. Some of the important properties are that the opposite angles are congruent, the sum of all the interior angles is $360^\circ $ and consecutive angles are supplementary.
Since $\left( {3x - 2} \right)$ and $\left( {x + 48} \right)$ are opposite angles of a parallelogram, we can equate these two expressions according to the above property.
$ \Rightarrow 3x - 2 = x + 48$
From this equation, we can find the unknown value of $'x'$
$ \Rightarrow 3x - 2 = x + 48 \Rightarrow 3x - x = 48 + 2 \Rightarrow 2x = 50$
We can divide both sides of the equation by $2$ to get the required value:
$ \Rightarrow x = \dfrac{{50}}{2} = 25$
Therefore, we got the value $x = 25$
So, the pair of opposite angles will be $\left( {3x - 2} \right) = 3 \times 25 - 2 = 75 - 2 = 73^\circ $ and $\left( {x + 48} \right) = 25 + 48 = 73^\circ $
Also, we know that the sum of all the angles is $360^\circ $ that is the sum of each of the pairs of the opposite angles is $360^\circ $.
$ \Rightarrow {\text{Sum of other pair of opposite angles}} = 360^\circ - \left( {73^\circ + 73^\circ } \right) = 360^\circ - 146^\circ = 214^\circ $
So each of these opposite angles will be $\dfrac{{214}}{2} = 107^\circ $
Thus we have a measure of all the four angles of the parallelogram.
Therefore, we get the measure of each of the angle is $73^\circ ,73^\circ ,107^\circ {\text{ and 107}}^\circ $

Note: In questions like this the geometric properties of different polygons plays a crucial role. An alternative approach can be to assume two remaining angles as some variable. Then use the property that states that the consecutive angles are supplementary in a parallelogram. Supplementary means that the sum of these angles will be equal to $180^\circ $ . So the other angle can be found by solving $180^\circ - 73^\circ = 107^\circ $