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If an angle of a parallelogram is two-third of its adjacent angle, find all the four angles.

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Last updated date: 14th Jun 2024
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Answer
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Hint: We are given a parallelogram and we are given one angle is some fraction of its adjacent angle and we have to find all the four angles. For this first we will assume the adjacent angle as any variable and then take the angle which is given as a fraction of adjacent angles. Applying the concept of parallelogram with the sum of adjacent angles in parallelogram is ${180^\circ}$. On equating the sum of angles to ${180^\circ}$ we will get the variable and on finding the variable from it we will find the two angles and by using the concept that opposite angles of parallelogram are equal will find all the four angles.

Complete step-by-step solution:
Step1: We are given a parallelogram in which an angle is two-third of its adjacent angles.
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Let’s take a parallelogram $ABCD$ in which let us assume $\angle A = x$ and $\angle B = \dfrac{{1x}}{3}$. As we know that the sum of adjacent angles is ${180^\circ}$. On forming the equation we get
$\angle A + \angle B = 180$
$x + \dfrac{{1x}}{3} = 180$
On adding the terms by taking L.C.M
$ \Rightarrow \dfrac{{3x + x}}{3} = 180$
$ \Rightarrow 4x = 180 \times 3$
On dividing by $4$
$x = \dfrac{{180 \times 3}}{4}$
On solving the equation we get
$x = {135^\circ}$
Putting the value of $x$ for $\angle B$ we get
$\angle B = \dfrac{1}{3} \times 135$
$\angle B = 45$
Step2: As we know that in parallelograms opposite angles are equal then $\angle B = \angle D$and $\angle A = \angle C$ as they form a pair of opposite angles. Hence $\angle B = \angle D = 45$ and $\angle A = \angle C = 135$

Hence four angles are $45,45,135,135$

Note: In such types of questions always draw a diagram by doing this it is quite helpful and in such questions there are small calculations only. These are properties based questions so remember and apply the concept properly. These do not require the long calculation just a right concept of properties. In such questions always take
Points to remember:
Sum of adjacent angles in parallelogram is $180$
Opposite angles are equal in a parallelogram