Question

The interior angles of a pentagon are in the ratio $2:3:4:5:6$ respectively, then what will be the sum of the first and the second angle?
A. ${90^0}$
B. ${140^0}$
C. ${135^0}$
D. ${180^0}$

Answer 1

Hint: We just have to generalize all the angles by multiplying them with a particular variable according to their ratio and equate their sum to the total sum of interior angles of a pentagon using the formula, Sum of interior angles of a regular pentagon $ = (n - 2) \times {180^0},$ where $n = 5$ for this equation i.e. number of sides of a regular pentagon due to which you will be able to find all the angles according to the generalized variable.

Complete step-by-step answer:
Let us suppose the angles of the pentagon are $2x,3x,4x,5x,$ and $6x$ according to the ratio given.
Let the number of sides of the pentagon be $n$
Now, we know that in a pentagon number of sides is equal to $n = 5$
We know that, in a regular pentagon, Sum of interior angles $ = (n - 2) \times {180^0}$
Substituting the value of $n = 5$ in the equation, we get
Sum of interior angles in a regular pentagon $ = (5 - 2) \times {180^0} = 3 \times {180^0} = {540^0}$
According to the question,
$2x + 3x + 4x + 5x + 6x = {540^0}$
$ \Rightarrow 20x = {540^0}$
$ \Rightarrow x = {27^0}$
Now, First angle $ = 2x = 2 \times {27^0} = {54^0}$ and Second angle $ = 3x = 3 \times {27^0} = {81^0}$
So, sum of the first and the second angle $ = {54^0} + {81^0} = {135^0}$
Hence, our required answer is ${135^0}$
Option C. is our correct answer

Note: Such type of questions can also be calculated by calculating the sum of the other three angles first and subtracting that from the ${540^0}$ due to which we will have to get the sum of the first and the second angle which we have been told to calculate.