Solving for the Number of Sides in a Polygon with Specific Interior Angles

In this guide, we will walk through the process of solving a problem involving the number of sides in a polygon given specific interior angles. We will explore how to apply geometric formulas and algebraic techniques to reach a solution. Understanding the steps involved will not only help in solving similar problems but also in comprehending the underlying principles of polygon geometry.

Problem Definition

We are given a polygon with n sides. Two of its interior angles are 150° and 160°, and the remaining angles are 170° each. We need to find the value of n.

Step-by-Step Solution

Using the Sum of Interior Angles Formula

The sum of the interior angles of a polygon can be determined using the formula:

$$text{Sum of interior angles} 180(n - 2) $$

In this problem, we have two angles measuring 150° and 160° and the remaining n - 2 angles each measure 170°. Therefore, we can write the total sum of these angles as:

$$ 150 160 (n - 2) cdot 170 $$

Equating the two expressions, we get:

$$ 180(n - 2) 150 160 (n - 2) cdot 170 $$

Let's simplify this equation step-by-step:

$$ 180(n - 2) 310 (n - 2) cdot 170 $$ $$ 180n - 360 310 170n - 340 $$ $$ 180n - 360 170n - 30 $$

Subtracting 170n from both sides:

$$ 10n - 360 -30 $$

Adding 360 to both sides:

$$ 10n 330 $$

Dividing both sides by 10:

$$ n 33 $$

Hence, the value of n is 33.

Alternative Method: Using Exterior Angles

For a more intuitive approach, we can use the fact that the exterior angles of a polygon must sum to 360°. Each exterior angle is the supplement of the corresponding interior angle. Therefore, the exterior angles in our case would be 30°, 20°, and 10°, respectively.

$$ text{Sum of exterior angles} 360 $$

Expressing the sum of the exterior angles as:

$$ 30 20 (n - 2) cdot 10 360 $$

Simplifying this expression:

$$ 50 10(n - 2) 360 $$ $$ 50 10n - 20 360 $$ $$ 10n 30 360 $$

Subtracting 30 from both sides:

$$ 10n 330 $$

Dividing both sides by 10:

$$ n 33 $$

This confirms that the value of n is 33.

Conclusion

Understanding the sum of interior angles and exterior angles of a polygon is crucial in solving such problems. The approach using the sum of interior angles directly gives us 180(n - 2) 150 160 170(n - 2), which simplifies to the result that the polygon has 33 sides. The alternative method using the sum of exterior angles further validates this conclusion.

Mastering these techniques will be beneficial for solving other geometric problems and for improving your overall understanding of polygon geometry.