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Answer. The number of sides is 5.
HINT. Use the fact that the sum of exterior angles in any convex polygon is 360 degrees.
Answer
Hint:This question is based on the sum of the internal angles of a polygon. First of all, we assume that the number of sides is $n$. Now by using the formula of the sum of internal angles of a polygon – Sum of internal angles of $n$sided polygon $=\left( n-2 \right)\times 180{}^\circ $By given angles in the question, we make an equation in $n$ and solve it and get the number of sides of a polygon.
Complete step by step answer:
Let us start by understanding what a polygon is. A polygon is a plane figure that is described by a finite number of straight-line segments called edges. There are an equal number of edges and angles in a polygon, and the sum of interior angles and side’s relation is – $S=\left( n-2 \right)\pi $ radian$S=\left( n-2 \right)\times 180{}^\circ $ degreeLet us assume that there are $n$ sides in a polygon. Thus, there are also $n$ internal angles in the polygon.According to the question, we have the data as below.Out of$n$, two angles are right angle. And the remaining $\left( n-2 \right)$ angles are $120{}^\circ $each.So, we can write that Sum of internal angles of polygon $=\left( 2\times 90{}^\circ \right)+\left( n-2 \right)120{}^\circ $ …(1)As we know that sum of interior angles of $n$ sided polygon$=\left( n-2 \right)\times 180{}^\circ $ …(2)Now, by equation (1) and (2), we can equate them as$\left( n-2 \right)\times 180{}^\circ =\left( 2\times 90{}^\circ \right)+\left( n-2 \right)120{}^\circ $On dividing by $60{}^\circ $on both sides, we get \[\Rightarrow \left( n-2 \right)\times \dfrac{180{}^\circ }{60}=\dfrac{\left( 2\times 90{}^\circ \right)}{60}+\left( n-2 \right)\dfrac{120{}^\circ }{60}\] $\Rightarrow 3\left( n-2 \right)=3+2\left( n-2 \right)$ By subtracting $2\left( n-2 \right)$ on both sides, we get$\Rightarrow 3\left( n-2 \right)-2\left( n-2 \right)=3+2\left( n-2 \right)-2\left( n-2 \right)$ $\Rightarrow \left( n-2 \right)=3$ By adding $2$ on both sides, we get$\Rightarrow n-2+2=3+2$ $\Rightarrow n=5$ Hence, the number of sides in the polygon is 5. Thus, the correct option is (a).A diagram of a pentagon ABCDE with two right angles and three angles each of $120{}^\circ $ is shown below.
Note:
Two angles of a polygon are right angles and the remaining are 120° each. Find the number of sides in it.
Let the number of sides = n
Sum of interior angles = (n - 2) × 180°
= 180n - 360°
Sum of 2 right angles = 2 × 90° = 180°
∴ Sum of other angles = 180n - 360° - 180°
= 180n - 540°
No.of vertices at which these angles are formed = n - 2
∴ Each interior angle = `(180"n" - 540)/("n" - 2)`
∴ `(180 "n" - 540)/("n" - 2) = 120°`
180n - 540 = 120n - 240
180n - 120n = - 240 + 540
60n = 300
n = `300/60`
n = 5
Concept: Sum of Angles of a Polynomial
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