Question

The measure of two adjacent angles of a parallelogram is in ratio 3:2. Find the measure of each of the angles of the parallelogram.

Answer 1

Hint: The opposite angles of the parallelogram are equal, and the total sum of all the angles of the parallelogram is equal to ${360^ \circ }$. Let the two adjacent angles of the parallelogram be $3x$ and $2x$ and solve for the $x$ using the mentioned properties of a parallelogram.

Complete step by step solution: It is given that the ratio of the two adjacent angles of the parallelogram is 3:2.
Let us consider the common factor of the ratio of the two adjacent angles of the parallelogram by $x$.
Thus the two adjacent angles of the parallelogram become $3x$ and $2x$.

It is known that the opposite angles of the parallelogram are equal. Therefore the other two angles of the parallelogram must be equal to the $3x$ and $2x$.
Thus the four angles of the parallelogram become $3x$, $2x$, $2x$ and $3x$.
It is also known that the total sum of the interior angles of any polygon with four sides is ${360^ \circ }$.
Thus the sum of $3x$, $2x$, $2x$ and $3x$ is equal to 360°.
$3x + 2x + 2x + 3x = {360^ \circ }$
Solving for $x$, we get,
$10x = {360^ \circ }$
Dividing the equation throughout by 10
$x = {36^ \circ }$
Substituting the value 36 for $x$ to find the angles of the parallelogram we get,
$2x = 2\left( {36} \right) = {72^ \circ }$ and $3x = 3\left( {36} \right) = {108^ \circ }$

Hence, the angles are ${108^ \circ },{72^ \circ },{108^ \circ },{72^ \circ }$

Note: Since the opposite angles of the parallelogram are equal and the total sum of all the four angles of the parallelogram is ${360^ \circ }$, the sum of the adjacent angles of the parallelogram must be equal to ${180^ \circ }$.