Exterior Angles Of Polygons: A Math Exploration

Hey there, math enthusiasts! Today, we're diving deep into the fascinating world of polygons, specifically focusing on the relationship between their exterior and interior angles. We'll tackle a classic geometry problem and uncover the secrets behind a special type of polygon. So, buckle up and get ready for a mathematical adventure!

Decoding the Angle Relationship

Let's start with the core question: What is the exterior angle of a regular polygon if it is two-thirds the interior angle? What name is given to the regular polygon? This might sound a bit intimidating at first, but don't worry, we'll break it down step by step.

To solve this, we need to understand the fundamental properties of polygons and their angles. Remember, a polygon is a closed, two-dimensional shape formed by straight line segments. A regular polygon is a special type where all sides and all angles are equal. This uniformity is key to our calculations.

Now, let's delve into the angles. Every interior angle of a polygon is formed inside the shape at a vertex (corner). An exterior angle, on the other hand, is formed by extending one side of the polygon and measuring the angle between the extended side and the adjacent side. A crucial relationship to remember is that the interior and exterior angles at any vertex of a polygon are supplementary, meaning they add up to 180 degrees. This is because they form a straight line together, and a straight line has an angle of 180 degrees. Keeping this supplementary angle relationship in mind is paramount to understanding the fundamental principle at play in solving geometry problems that deal with angles. For example, in a square, each interior angle is 90 degrees, and each exterior angle is also 90 degrees. The sum of these two angles is 180 degrees, satisfying the supplementary angle relationship. Similarly, in an equilateral triangle, each interior angle is 60 degrees, and each exterior angle is 120 degrees. The sum of 60 and 120 is again 180 degrees, demonstrating the consistency of this principle. As polygons become more complex, with more sides and angles, this relationship remains constant. In a regular hexagon, each interior angle measures 120 degrees, and each exterior angle measures 60 degrees, still summing up to 180 degrees. Therefore, whether dealing with simple shapes like triangles and squares or more complex polygons like decagons and dodecagons, the interior and exterior angles at any vertex will always add up to 180 degrees, providing a reliable foundation for calculating unknown angles and understanding polygon properties.

With this foundation in place, we can start translating the problem into mathematical terms. Let's represent the interior angle as 'i' and the exterior angle as 'e'. According to the problem, the exterior angle (e) is two-thirds of the interior angle (i). We can write this as an equation:

e = (2/3) * i

We also know that the interior and exterior angles are supplementary:

i + e = 180

Now we have a system of two equations with two variables, which we can solve to find the values of 'i' and 'e'. This is a classic algebraic setup, and by solving it, we can unravel the mystery of the angles in our polygon. To solve this system, we can use substitution. From the first equation, we know that e = (2/3) * i. We can substitute this expression for 'e' into the second equation:

i + (2/3) * i = 180

Now, we have an equation with only one variable, 'i'. We can combine the terms on the left side of the equation. Think of 'i' as 1 * i. So, we have:

(1 + 2/3) * i = 180

To add 1 and 2/3, we need to express 1 as a fraction with a denominator of 3, which is 3/3. So, we have:

(3/3 + 2/3) * i = 180

Adding the fractions, we get:

(5/3) * i = 180

To isolate 'i', we need to multiply both sides of the equation by the reciprocal of 5/3, which is 3/5:

(3/5) * (5/3) * i = 180 * (3/5)

The (3/5) and (5/3) on the left side cancel each other out, leaving us with:

i = 180 * (3/5)

Now we can multiply 180 by 3/5. First, we multiply 180 by 3:

180 * 3 = 540

Then, we divide 540 by 5:

540 / 5 = 108

So, the interior angle 'i' is 108 degrees. This step-by-step algebraic manipulation shows how we can start with the initial equations and use substitution and arithmetic operations to isolate and solve for 'i'. This method is a fundamental skill in solving mathematical problems and highlights how seemingly complex problems can be simplified through a systematic approach. Once we find the value of 'i', we're one step closer to fully understanding the properties of our polygon.

Now that we've found the interior angle, we can substitute this value back into either of our original equations to solve for the exterior angle 'e'. Let's use the equation:

e = (2/3) * i

We know that i = 108 degrees, so we substitute that into the equation:

e = (2/3) * 108

To multiply 108 by 2/3, we can first multiply 108 by 2:

108 * 2 = 216

Then, we divide 216 by 3:

216 / 3 = 72

So, the exterior angle 'e' is 72 degrees. This calculation not only gives us the value of 'e' but also reinforces the practicality of using the equations we set up earlier. By substituting the known value of 'i' into the equation e = (2/3) * i, we efficiently determined the measure of the exterior angle. The careful execution of arithmetic operations ensures that we arrive at the correct solution. The fact that the exterior angle is 72 degrees now gives us a clearer picture of the polygon we're dealing with, as it provides a key piece of information for identifying the specific type of regular polygon that has these angle measurements. With both the interior and exterior angles known, we are well-equipped to move on to the final stage of identifying the polygon itself.

Unmasking the Polygon's Identity

We've determined that the interior angle of our regular polygon is 108 degrees, and the exterior angle is 72 degrees. But what does this tell us about the polygon itself? To figure this out, we need to recall another important property of polygons: the sum of the exterior angles of any polygon, regardless of the number of sides, is always 360 degrees. This is a powerful fact that we can use to find the number of sides.

Since we know each exterior angle is 72 degrees and the total exterior angle sum is 360 degrees, we can divide 360 by 72 to find the number of sides:

Number of sides = 360 / 72 = 5

Therefore, our regular polygon has 5 sides. Now, do you remember what a 5-sided polygon is called? That's right, it's a pentagon! And since all sides and angles are equal, it's a regular pentagon. The calculation of the number of sides using the exterior angle sum is a neat trick in geometry. The fact that the sum of exterior angles is always 360 degrees is a consistent property that applies to all polygons, convex or concave, regular or irregular. This consistency makes it a reliable tool for finding the number of sides when the measure of each exterior angle is known. By dividing the total sum of 360 degrees by the measure of one exterior angle, we directly obtain the number of angles, which is equal to the number of sides in the polygon. This method bypasses the need to know the interior angles or use other more complicated formulas, making it an efficient way to identify polygons, especially in cases where the exterior angle is easily determined.

Why This Matters: Real-World Connections

You might be wondering,