Question

The angles of the pentagon are in ratio \[4:8:6:4:5\]. Find the greatest angle of the pentagon.
A. \[120^\circ \]
B. \[140^\circ \]
C. \[160^\circ \]
D. \[150^\circ \]

Answer 1

Hint: First we first assume that the angles of a pentagon are \[4x\], \[8x\], \[6x\], \[4x\] and \[5x\]. Then we will use the angle sum property of the polygon that the sum of the interior angle of a polygon is \[180^\circ \left( {n - 2} \right)\], where \[n\] is the number of sides of the polygon. Then we will substitute the obtained value of \[x\] in the angles of the triangle to find the largest angle.

Complete step by step answer:

We are given that the angles of a pentagon are in the ratio \[4:8:6:4:5\].
Let us assume that the angles of a pentagon are \[4x\], \[8x\], \[6x\], \[4x\] and \[5x\].
We know the angle sum property of the polygon that the sum of the interior angle of a polygon is \[180^\circ \left( {n - 2} \right)\], where \[n\] is the number of sides of the polygon.
Replacing 5 for \[n\] in the above formula to find the sum of the interior angle of a pentagon, we get
\[
   \Rightarrow 180^\circ \left( {5 - 2} \right) \\
   \Rightarrow 180^\circ \times 3 \\
   \Rightarrow 540^\circ \\
 \]
Then, we have
\[
   \Rightarrow 4x + 8x + 6x + 4x + 5x = 540^\circ \\
   \Rightarrow 27x = 540^\circ \\
 \]
Dividing the above equation by 27 on both sides, we get
\[
   \Rightarrow \dfrac{{27x}}{{27}} = \dfrac{{540^\circ }}{{27}} \\
   \Rightarrow x = 20^\circ \\
 \]
Substituting the above value of \[x\] in the five angles of the triangle, we get
\[ \Rightarrow 4\left( {20^\circ } \right) = 80^\circ \]
\[ \Rightarrow 8\left( {20^\circ } \right) = 160^\circ \]
\[ \Rightarrow 6\left( {20^\circ } \right) = 120^\circ \]
\[ \Rightarrow 4\left( {20^\circ } \right) = 80^\circ \]
\[ \Rightarrow 5\left( {20^\circ } \right) = 100^\circ \]
Thus, the greatest angle of the pentagon is \[160^\circ \].
Hence, option C is correct.

Note: In solving this question, we have multiplied the given ratio with some unknown variable. Then use the fact that the sum of angle of triangle is \[180^\circ \left( {n - 2} \right)\], where \[n\] is the number of sides to obtain a linear equation. Then this question is really simple to solve.
We can also find the greatest angle by taking \[x\] is a constant and the number, which is greater on ratio will be the greatest ratio 8, so multiplying \[x\] with 8, we get
\[8\left( {20^\circ } \right) = 160^\circ \]
Thus, the greatest angle of the pentagon is \[160^\circ \].