Question

The measure of two consecutive angles in a parallelogram are in the ratio $4:11$. What is the measure of the four angles of the parallelogram?

Answer 1

Hint:We know that the sum of two adjacent angles in a parallelogram is ${180^ \circ }$. So, with the help of this property, we can find these two angles as the ratio of these two angles is given in the question. We also know that opposite sides of a parallelogram are parallel. So, with the help of this property, we can find other two angles also.

Complete step by step answer:

In the above question, it is given that we have a parallelogram in which the ratio of its consecutive angles is given as $4:11$.
Let the value of $\angle A$ is $4x$.
Therefore, the value of $\angle B$ will be $11x$.
We know that the sum of consecutive angles in a parallelogram is $180$ degrees.
Therefore,
$ \Rightarrow 4x + 11x = 180$
$ \Rightarrow 15x = 180$
On cross-multiplication, we get
$ \Rightarrow x = \dfrac{{180}}{{15}}$
On division, we get
$ \Rightarrow x = 12$
Now, $\angle A = 4x = 4 \times 12 = 48$
$\angle B = 11x = 11 \times 12 = 132$
Also, we know that opposite sides and opposite angles in a parallelogram are equal.
hence, $\angle A = \angle C\,\,and\,\,\angle B = \angle D$
Therefore, $\angle C = {48^ \circ }\,and\,\,\angle D = {132^ \circ }$

Note:We can also do this question by adding all the angles and putting their sum up to $360$ degrees. The properties of a parallelogram are as follows: The opposite sides are parallel and congruent. The opposite angles are congruent. The consecutive angles are supplementary. If any of the angles is a right angle, then all the other angles will be the right angle. The two diagonals bisect each other.