Solving For X In A Regular Pentagon Exterior Angle Problem

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Let's embark on a fascinating journey into the world of geometry, where we'll explore the properties of regular pentagons and their exterior angles. Our mission is to decipher the value of 'x' in a specific scenario, where an exterior angle of a regular pentagon measures $(2x)^{\circ}$. To achieve this, we'll delve into the fundamental concepts of polygons, exterior angles, and the unique characteristics of regular pentagons. By carefully dissecting the problem and applying the relevant geometric principles, we'll arrive at the solution and gain a deeper understanding of these mathematical concepts.

Understanding Regular Pentagons and Exterior Angles

Before we dive into solving for 'x', it's crucial to establish a solid foundation by understanding the key concepts involved. A pentagon is a polygon, a closed two-dimensional shape, with five sides and five angles. A regular pentagon is a special type of pentagon where all five sides are of equal length, and all five interior angles are equal in measure. This symmetry and uniformity are the defining characteristics of a regular pentagon.

Now, let's turn our attention to exterior angles. An exterior angle of a polygon is formed by extending one side of the polygon and measuring the angle between the extended side and the adjacent side. Imagine walking along the perimeter of a polygon; each time you turn at a corner, you're essentially forming 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 the polygon has. This principle will be instrumental in our quest to find the value of 'x'.

In the context of our regular pentagon, since it has five sides and five vertices, it also has five exterior angles. Because it's a regular pentagon, all these exterior angles are equal in measure. This uniformity simplifies our task significantly, as we can divide the total exterior angle sum (360 degrees) by the number of angles (5) to find the measure of each individual exterior angle. This measure will then be related to the expression $(2x)^{\circ}$, allowing us to solve for 'x'. Let's move on to the next step where we calculate the measure of an exterior angle of a regular pentagon.

Calculating the Measure of an Exterior Angle

As we established earlier, the sum of the exterior angles of any polygon, including a pentagon, is always 360 degrees. In the case of a regular pentagon, all five exterior angles are congruent, meaning they have the same measure. To find the measure of a single exterior angle, we simply divide the total sum of exterior angles (360 degrees) by the number of exterior angles (which is equal to the number of sides, 5).

So, the measure of one exterior angle of a regular pentagon is calculated as follows:

Measure of one exterior angle=Total sum of exterior anglesNumber of exterior angles=360∘5=72∘\text{Measure of one exterior angle} = \frac{\text{Total sum of exterior angles}}{\text{Number of exterior angles}} = \frac{360^{\circ}}{5} = 72^{\circ}

This calculation reveals a crucial piece of information: each exterior angle of a regular pentagon measures 72 degrees. Now, we can connect this value to the given expression, $(2x)^{\circ}$, which represents the measure of one exterior angle in our problem. By setting these two values equal to each other, we form an equation that we can solve for 'x'. This is the key step in unlocking the solution, as it bridges the gap between the geometric property of the pentagon and the algebraic expression involving 'x'. Let's proceed to set up and solve this equation in the next section.

Setting Up and Solving the Equation for x

Now that we know the measure of one exterior angle of a regular pentagon is 72 degrees, and we're given that this angle is also represented by $(2x)^{\circ}$, we can set up a simple equation to solve for 'x'. The equation is:

2x=722x = 72

This equation states that twice the value of 'x' is equal to 72. To isolate 'x' and find its value, we need to perform a basic algebraic operation: dividing both sides of the equation by 2. This will undo the multiplication by 2 on the left side and leave 'x' by itself.

Performing the division, we get:

2x2=722\frac{2x}{2} = \frac{72}{2}

x=36x = 36

Therefore, the value of 'x' that satisfies the given condition is 36. This result aligns perfectly with the properties of regular pentagons and exterior angles, confirming the validity of our solution. We've successfully navigated through the problem, applying geometric principles and algebraic techniques to arrive at the answer. Let's solidify our understanding by reviewing the steps we took and highlighting the key concepts involved.

Reviewing the Solution and Key Concepts

Let's recap the journey we undertook to find the value of 'x'. We started with the problem statement, which informed us that one exterior angle of a regular pentagon has a measure of $(2x)^{\circ}$. Our goal was to determine the value of 'x'.

To achieve this, we first established a solid understanding of the fundamental concepts. We defined a pentagon and, more specifically, a regular pentagon, emphasizing its equal sides and equal angles. We then explored the concept of exterior angles, recognizing that they are formed by extending one side of a polygon and measuring the angle between the extension and the adjacent side. A critical point we highlighted was that the sum of the exterior angles of any polygon is always 360 degrees.

Next, we applied this knowledge to our regular pentagon. Since it has five equal exterior angles, we calculated the measure of one exterior angle by dividing the total exterior angle sum (360 degrees) by the number of angles (5), resulting in 72 degrees. This was a pivotal step, as it allowed us to connect the geometric property of the pentagon to the algebraic expression involving 'x'.

We then set up the equation $2x = 72$, equating the calculated exterior angle measure (72 degrees) with the given expression $(2x)^{\circ}$. Solving this equation involved a simple algebraic step: dividing both sides by 2, which yielded the solution $x = 36$.

In conclusion, by understanding the properties of regular pentagons and exterior angles, and by applying basic algebraic techniques, we successfully determined that the value of 'x' is 36. This problem serves as a great example of how geometry and algebra can work together to solve mathematical puzzles. The answer is **D. x=36.

Additional Practice Problems

To further solidify your understanding of regular pentagons and exterior angles, consider tackling these additional practice problems:

  1. What is the measure of each interior angle of a regular pentagon?
  2. If an exterior angle of a regular polygon measures 45 degrees, how many sides does the polygon have?
  3. The sum of the interior angles of a polygon is 1440 degrees. How many sides does the polygon have? What is the measure of each exterior angle if the polygon is regular?

By working through these problems, you'll strengthen your grasp of the concepts and enhance your problem-solving skills in geometry.