Question

Three angles of a quadrilateral are in the ratio 3: 5: 8. The mean of these angles is ${{80}^{\circ }}$. Find all the four angles of the quadrilateral.

Answer 1

Hint: Here, since angles are in the ratio 3: 5: 8, we can take the angles as $3x,5x$ and $8x$. Next, with the help of formula of mean, sum of the values divided by the total number of values we can calculate the value of $x$. Now, with the help of $x$ we can find three angles. To find the fourth angle we have to use the theorem that the sum of four angles of the quadrilateral is ${{360}^{\circ }}$. Now, by solving we will get the fourth angle.

Complete step-by-step solution -
Here, for a better understanding, let us draw the figure.

We are given that three angles of a quadrilateral are in the ratio 3: 5: 8.
Therefore, let us take the three angles as $3x,5x$ and $8x$respectively.
Here, it is also given that the mean of these angles is ${{80}^{\circ }}$.
We know by the definition of mean that:
$mean=\dfrac{sum\text{ }of\text{ }the\text{ }values}{total\text{ }number\text{ }of\text{ }values}$
$mean=\dfrac{3x+5x+8x}{3}$
$80=\dfrac{16x}{3}$ … (given that mean = 80)
Therefore, by cross multiplication, we get the equation:
$\begin{align}
  & 80\times 3=16x \\
 & 240=16x \\
\end{align}$
Now, by taking 16 to the left side we get:
$\begin{align}
  & \dfrac{240}{16}=x \\
 & 15=x \\
\end{align}$
Therefore, we got the value of $x=15$.
Now we have to find the three angles of the quadrilateral. i.e. $3x,5x$ and $8x$. i.e.
$\begin{align}
  & 3x=3\times 15=45 \\
 & 5x=5\times 15=75 \\
 & 8x=8\times 15=120 \\
\end{align}$
Hence, the three angles of the quadrilateral are ${{45}^{\circ }},{{75}^{\circ }}$ and ${{120}^{\circ }}$.
Next, we have to find the fourth angle of the quadrilateral.
We know by a theorem that the sum of the four angles of a quadrilateral is ${{360}^{\circ }}$.
From the figure we can say that the unknown angle is A. Now, we can find the value of A with the help of the above theorem. Therefore, we can write:
$\begin{align}
  & A+{{45}^{\circ }}+{{75}^{\circ }}+{{120}^{\circ }}={{360}^{\circ }} \\
 & A+{{240}^{\circ }}={{360}^{\circ }} \\
\end{align}$
By taking ${{240}^{\circ }}$ to the right side, it becomes -${{240}^{\circ }}$. Then our equation becomes:
$\begin{align}
  & A={{360}^{\circ }}-{{240}^{\circ }} \\
 & A={{120}^{\circ }} \\
\end{align}$
Hence, we got the fourth angle of the quadrilateral as ${{120}^{\circ }}$.
Therefore, all four angles of the quadrilateral are ${{45}^{\circ }},{{75}^{\circ }},{{120}^{\circ }}$ and ${{120}^{\circ }}$ respectively.

Note: Here, we have to find the angles of the quadrilateral. A quadrilateral is made up of two triangles. We know that the sum of the angles of a triangle is ${{180}^{\circ }}$. Therefore, the sum of two triangles will be ${{360}^{\circ }}$ which will be the sum of the angles of the quadrilateral. So if you know three angles, the fourth one you can find using this theorem.