Decagon Exterior Angle Problem Solving For X

This article delves into the fascinating world of polygons, specifically focusing on regular decagons and their exterior angles. We'll unravel the mystery behind the equation (3x + 6)° and embark on a step-by-step journey to determine the value of 'x.' Whether you're a student grappling with geometry or simply a math enthusiast, this exploration will enhance your understanding of geometric principles and problem-solving techniques.

Understanding Regular Decagons and Their Exterior Angles

To solve this problem effectively, it's crucial to establish a strong foundation in the properties of regular decagons and their exterior angles. A decagon, by definition, is a polygon with ten sides and ten angles. When we refer to a regular decagon, we're emphasizing that all its sides are of equal length, and all its interior angles are congruent (equal in measure). This regularity is key to unlocking the solution.

Now, let's turn our attention to exterior angles. Imagine extending one side of the decagon outward. The angle formed between this extended side and the adjacent side is an exterior angle. At each vertex of the decagon, we can form an exterior angle. A fundamental property of polygons is that the sum of all exterior angles, one at each vertex, always equals 360 degrees, regardless of the number of sides. This holds true for triangles, quadrilaterals, pentagons, and, of course, decagons. Understanding this property is paramount to finding the value of 'x'. In the case of a regular polygon, where all sides and angles are equal, all exterior angles are also equal in measure. This simplifies our task considerably. To find the measure of a single exterior angle in a regular decagon, we simply divide the total sum of exterior angles (360 degrees) by the number of sides (10). This gives us 36 degrees per exterior angle. This foundational knowledge sets the stage for us to tackle the given equation and solve for 'x'.

Setting Up the Equation: Connecting the Given Information

The problem statement presents us with a crucial piece of information: each exterior angle of the regular decagon has a measure of (3x + 6) degrees. We've already established that each exterior angle of a regular decagon measures 36 degrees. Now, we can bridge the gap between these two pieces of information by setting up an equation. The expression (3x + 6) represents the measure of each exterior angle, and we know this measure is also equal to 36 degrees. Therefore, we can write the equation as:

3x + 6 = 36

This equation is the heart of our problem. It mathematically expresses the relationship between the unknown variable 'x' and the known measure of the exterior angle. By solving this equation, we'll unveil the value of 'x'. The equation is a linear equation, meaning the highest power of the variable 'x' is 1. Linear equations are relatively straightforward to solve, typically involving isolating the variable on one side of the equation. This step involves using algebraic manipulations to maintain the equality while moving terms around. Before we proceed with solving the equation, let's pause and appreciate the power of this simple equation. It encapsulates the geometric properties of a decagon and translates them into an algebraic expression, allowing us to use the tools of algebra to find a specific value. This connection between geometry and algebra is a recurring theme in mathematics, highlighting the interconnectedness of different mathematical concepts. Now, with our equation firmly in place, we're ready to embark on the final stage: solving for 'x'.

Solving for x: A Step-by-Step Approach

Now that we have the equation 3x + 6 = 36, we can proceed to solve for 'x'. The goal is to isolate 'x' on one side of the equation. This involves performing algebraic operations on both sides to maintain the equality. The first step in isolating 'x' is to eliminate the constant term (+6) from the left side of the equation. To do this, we subtract 6 from both sides of the equation:

3x + 6 - 6 = 36 - 6

This simplifies to:

3x = 30

Now, we have 3x equal to 30. To isolate 'x' completely, we need to get rid of the coefficient 3. Since 'x' is being multiplied by 3, we perform the inverse operation: division. We divide both sides of the equation by 3:

(3x) / 3 = 30 / 3

This simplifies to:

x = 10

Therefore, the value of 'x' that satisfies the equation is 10. This means that when x is equal to 10, the expression (3x + 6) will indeed equal the measure of an exterior angle of a regular decagon, which is 36 degrees. We have successfully navigated the algebraic steps to find the solution. However, it's always prudent to verify our answer to ensure accuracy. We can substitute x = 10 back into the original equation to check if it holds true. This verification step adds confidence to our solution and reinforces our understanding of the problem-solving process. In the next section, we'll verify our solution and discuss the implications of our findings.

Verification and Conclusion: Confirming the Solution

To ensure the accuracy of our solution, we can substitute the value of x we found (x = 10) back into the original expression for the exterior angle, (3x + 6). This process is called verification and is a crucial step in problem-solving.

Substituting x = 10 into (3x + 6), we get:

3(10) + 6 = 30 + 6 = 36

The result, 36 degrees, matches the measure of each exterior angle of a regular decagon, which we calculated earlier. This confirms that our solution, x = 10, is correct. The verification step not only validates our answer but also reinforces our understanding of the problem and the steps we took to solve it. It demonstrates the logical flow of our reasoning and the consistency of our calculations.

In conclusion, we have successfully determined the value of 'x' in the given problem. By understanding the properties of regular decagons, particularly the sum of their exterior angles, we were able to set up an equation relating the given expression (3x + 6) to the actual measure of an exterior angle. Solving this equation using algebraic techniques, we found that x = 10. The verification step further solidified our confidence in the solution. This problem exemplifies the interplay between geometry and algebra, showcasing how algebraic tools can be used to solve geometric problems. The ability to translate geometric concepts into algebraic expressions and vice versa is a valuable skill in mathematics and its applications. We hope this detailed exploration has provided a clear understanding of the problem-solving process and has enhanced your appreciation for the beauty and interconnectedness of mathematics.

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