Finding Angle Measures: A Geometry Problem

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Hey guys! Let's dive into a cool geometry problem today that involves angles, complements, and supplements. We're going to break down the problem step by step, so it's super easy to understand. If you've ever struggled with geometry, don't worry; we've got your back! We’ll explore how to find the measure of an angle when given a relationship between its complement and supplement. So, grab your pencils and let’s get started!

Understanding Complementary and Supplementary Angles

Before we even attempt to solve this problem, it’s super important that we understand the basic concepts of complementary and supplementary angles. These are fundamental building blocks in geometry, and knowing them inside and out will make solving this and similar problems a breeze. Let’s break it down in a way that’s easy to remember and apply.

What are Complementary Angles?

Complementary angles are basically two angles that, when you add their measures together, give you a total of 90 degrees. Think of it as completing a right angle! If you've got an angle that’s, say, 30 degrees, its complement would be an angle that adds up with it to make 90 degrees. So, 90 - 30 = 60 degrees. Therefore, the complement of a 30-degree angle is a 60-degree angle. Easy peasy, right? Whenever you see the word “complement,” remember 90 degrees – that’s your magic number. You'll often see this in right triangles, where the two acute angles are complementary.

What are Supplementary Angles?

Now, let’s talk about supplementary angles. These are two angles whose measures add up to 180 degrees. Think of a straight line or a flat angle – that’s 180 degrees. So, if you have an angle of 70 degrees, its supplement is the angle you need to add to 70 to get 180. That would be 180 - 70 = 110 degrees. So, the supplement of a 70-degree angle is 110 degrees. Just like with complementary angles, there’s a handy number to remember here: 180. Whenever “supplement” pops up, 180 should ring a bell in your mind. This concept is super useful when you’re dealing with angles on a straight line or in various polygons.

Why Are These Concepts Important?

Grasping these concepts is key because they pop up all the time in geometry problems. Knowing what complementary and supplementary angles are allows you to set up equations and solve for unknown angles. They're like the secret ingredients for cracking geometric puzzles! Without a solid understanding of complements and supplements, tackling problems like the one we're about to solve would be much, much harder. Think of it this way: they are the alphabet of geometry, and once you know them, you can start forming words and sentences (or, in this case, solve problems and theorems!). So, make sure you've got these down pat before moving on.

Setting Up the Equation

Okay, now that we've refreshed our understanding of complementary and supplementary angles, let's get back to the problem at hand. The question states: “Find the measure of an angle if its complement is one-fourth of its supplement.” To solve this, the first thing we need to do is translate this word problem into a mathematical equation. Don’t worry; it’s not as scary as it sounds! We just need to break it down piece by piece and turn each part into its mathematical equivalent. So, let’s roll up our sleeves and get started!

Defining Our Variable

First things first, we need to define our variable. In this case, we’re trying to find the measure of an angle, so let’s call that angle “x.” This is a crucial first step because it gives us something to work with. The angle we’re looking for is our unknown, and “x” is going to represent that unknown in our equation. So, whenever we refer to “x,” we’re talking about the angle whose measure we want to find. This simple step makes the rest of the process much clearer and more manageable.

Expressing the Complement

Next, we need to express the complement of angle x. Remember, the complement of an angle is what you add to it to get 90 degrees. So, the complement of angle x would be 90 - x. Think of it as subtracting the angle from 90 to find its complementary angle. This expression, 90 - x, is going to be a key component of our equation. It represents the angle that, when added to x, equals 90 degrees. Keep this in mind as we move forward – it’s going to be an important part of our puzzle.

Expressing the Supplement

Now, let’s tackle the supplement of angle x. The supplement of an angle is what you add to it to get 180 degrees. So, the supplement of angle x would be 180 - x. Just like with the complement, we’re subtracting the angle from the total (in this case, 180) to find its supplementary angle. This expression, 180 - x, is the other crucial component we need to form our equation. It represents the angle that, when added to x, equals 180 degrees. With both the complement and supplement expressed in terms of x, we’re one big step closer to setting up our equation.

Forming the Equation

Now comes the fun part: putting it all together! The problem tells us that the measure of the complement is one-fourth of the measure of the supplement. In mathematical terms, this translates to:

90 - x = (1/4) * (180 - x)

This equation is the heart of our problem. It states that the complement of x (90 - x) is equal to one-fourth (1/4) of the supplement of x (180 - x). By setting up this equation correctly, we’ve laid the groundwork for solving the problem. All that’s left is to solve for x, which we’ll tackle in the next section. Give yourself a pat on the back for making it this far – the toughest part is often translating the words into math!

Solving the Equation

Alright, we’ve successfully set up our equation: 90 - x = (1/4) * (180 - x). Now, it’s time to roll up our sleeves and solve for x. Don’t worry; we’ll take it one step at a time, making sure each step is clear and easy to follow. Solving equations is a fundamental skill in algebra, and it's super satisfying when you nail it! So, let’s dive in and get that value for x.

Distribute the 1/4

The first thing we want to do is get rid of that fraction. To do this, we’ll distribute the 1/4 across the terms inside the parentheses on the right side of the equation. So, (1/4) * (180 - x) becomes (1/4) * 180 - (1/4) * x. Let’s simplify that a bit:

(1/4) * 180 = 45

(1/4) * x = x/4

So, our equation now looks like this:

90 - x = 45 - x/4

We’ve managed to clear the parentheses and simplify the right side. That’s one less thing to worry about! Distributing the fraction is a key step in making the equation more manageable. Now, we can move on to getting all the x terms on one side and the constants on the other.

Get Rid of the Fraction

To make things even easier, let’s get rid of that remaining fraction, x/4. The easiest way to do this is to multiply every term in the equation by 4. This will eliminate the denominator and give us a cleaner equation to work with. So, let’s multiply both sides by 4:

4 * (90 - x) = 4 * (45 - x/4)

Now, distribute the 4 on both sides:

4 * 90 - 4 * x = 4 * 45 - 4 * (x/4)

This simplifies to:

360 - 4x = 180 - x

Awesome! We’ve successfully eliminated the fraction. Multiplying by the denominator is a trick that comes in handy in many algebraic equations. Now that we have a fraction-free equation, we can rearrange the terms more easily.

Isolate the x Terms

Now, let’s gather all the x terms on one side of the equation. We’ll add 4x to both sides to get rid of the -4x on the left side:

360 - 4x + 4x = 180 - x + 4x

This simplifies to:

360 = 180 + 3x

Great! We’ve got all our x terms on the right side. Now, let’s move the constant terms to the other side so we can isolate the x term completely.

Isolate the Constant Terms

To get the constant terms on one side, we’ll subtract 180 from both sides of the equation:

360 - 180 = 180 + 3x - 180

This simplifies to:

180 = 3x

We’re almost there! We’ve isolated the x term on one side and the constant on the other. Now, all that’s left is to solve for x.

Solve for x

Finally, to solve for x, we need to divide both sides of the equation by 3:

180 / 3 = 3x / 3

This simplifies to:

x = 60

Woohoo! We did it! We found that x equals 60. That means the measure of the angle we were looking for is 60 degrees. Give yourself a big pat on the back – you’ve successfully solved a geometry problem by setting up and solving an equation. Now, let’s make sure our answer makes sense in the context of the original problem.

Checking the Solution

Okay, we've found that the measure of the angle, x, is 60 degrees. But, before we declare victory, it’s super important to check our solution. This step is like the quality control of math problems – it ensures that our answer not only makes sense mathematically but also fits the conditions of the original problem. So, let’s put our answer to the test and make sure everything checks out!

Calculate the Complement

First, let’s calculate the complement of our angle. Remember, the complement is what we add to the angle to get 90 degrees. So, the complement of a 60-degree angle is:

90 - 60 = 30 degrees

So, the complement of our angle is 30 degrees. Got it! Now, let’s move on to the supplement.

Calculate the Supplement

Next up, let’s find the supplement of our angle. The supplement is what we add to the angle to get 180 degrees. So, the supplement of a 60-degree angle is:

180 - 60 = 120 degrees

Alright, the supplement of our angle is 120 degrees. We’re making good progress! Now, we need to check if these values fit the condition given in the problem.

Verify the Condition

The problem stated that the measure of the complement is one-fourth of the measure of the supplement. Let’s see if this holds true for our solution. We need to check if:

30 = (1/4) * 120

Let’s calculate (1/4) * 120:

(1/4) * 120 = 30

Yes! Our equation holds true. The complement (30 degrees) is indeed one-fourth of the supplement (120 degrees). This confirms that our solution is correct. We’ve not only solved the problem but also verified our answer. Pat yourself on the back once more – you’ve earned it!

Final Answer

So, to wrap it all up, we've successfully found the measure of the angle. The final answer is:

x = 60 degrees

Conclusion

And there you have it! We've successfully navigated a geometry problem involving complementary and supplementary angles. Guys, remember, the key is to break down the problem into manageable steps, translate the words into mathematical expressions, set up the equation, solve for the unknown, and most importantly, check your solution. Geometry might seem tricky at first, but with practice and a clear understanding of the basics, you can totally nail it. Keep practicing, and you’ll become a geometry whiz in no time! Well done, and keep up the awesome work!