Finding Angle ABC: A Step-by-Step Math Guide

by Dimemap Team 45 views

Hey math enthusiasts! Let's dive into a geometry problem that's super common and important to understand. We're going to break down how to find the measure of angle ABC when given some specific relationships with angle DEF. This isn't just about getting the right answer; it's about understanding the concepts of supplementary angles and algebraic manipulation. So, grab your pencils, and let's get started. We'll explore the problem, break down the solution step-by-step, and make sure you understand the underlying concepts so you can ace similar problems in the future. Ready? Let's go!

Understanding the Problem: Supplementary Angles

Supplementary angles are the key to unlocking this problem. When two angles are supplementary, it means they add up to 180 degrees. Think of it like a straight line – a straight line forms an angle of 180 degrees, and if you split that line into two parts, you create two supplementary angles. In our case, we're told that angle ABC and angle DEF are supplementary. This is the first crucial piece of information we'll use. Mathematically, we can represent this relationship as: m(ABC^)+m(DEF^)=180om(\widehat{ABC}) + m(\widehat{DEF}) = 180^ {o}. Where m(ABC^)m(\widehat{ABC}) represents the measure of angle ABC and m(DEF^)m(\widehat{DEF}) represents the measure of angle DEF. Now, that's just a fancy way of saying: the angle ABC plus the angle DEF equals 180 degrees. Easy peasy, right?

But wait, there's more! The problem also tells us about the relationship between the two angles. Specifically, it states that the measure of angle ABC is 4 times the measure of angle DEF, minus 20 degrees. This is where we need to bring in a little bit of algebra to represent this relationship. We can write this as: m(ABC^)=4∗m(DEF^)−20om(\widehat{ABC}) = 4 {*} m(\widehat{DEF}) - 20^ {o}.

This equation tells us that if you take the angle DEF, multiply it by four, and then subtract 20, you'll get the measure of angle ABC. Understanding this relationship is vital because it allows us to create an equation with only one unknown variable. The core of this problem lies in understanding these two pieces of information: the supplementary nature of the angles and the relationship between their measures. With these two relationships, we can set up and solve the problem. The first step towards a solution is identifying what is known, and what is unknown, then applying the definitions and relationships of angles. Let's move on to the next section to see how we can solve this problem step by step!

Step-by-Step Solution: Finding the Measure of Angle ABC

Alright, guys, now comes the fun part: solving the problem! We've got two equations and two unknowns, which is perfect for solving. Here's how we'll do it, step by step, so you won't get lost along the way. First, we know that m(ABC^)+m(DEF^)=180om(\widehat{ABC}) + m(\widehat{DEF}) = 180^ {o} and m(ABC^)=4∗m(DEF^)−20om(\widehat{ABC}) = 4 {*} m(\widehat{DEF}) - 20^ {o}. We can use the second equation and substitute the value of m(ABC^)m(\widehat{ABC}) in the first equation. This is a common technique in algebra called substitution. So, instead of writing m(ABC^)m(\widehat{ABC}), we'll replace it with what it equals, which is 4∗m(DEF^)−20o4 {*} m(\widehat{DEF}) - 20^ {o}. This gives us:

(4∗m(DEF^)−20o)+m(DEF^)=180o(4 {*} m(\widehat{DEF}) - 20^ {o}) + m(\widehat{DEF}) = 180^ {o}.

See how we've replaced one unknown with an expression that relates it to the other unknown? Now we can simplify this equation. Combining like terms (the m(DEF^)m(\widehat{DEF}) terms), we get: 5∗m(DEF^)−20o=180o5 {*} m(\widehat{DEF}) - 20^ {o} = 180^ {o}. Next, let's get the term with m(DEF^)m(\widehat{DEF}) by itself. To do this, we need to get rid of the -20 degrees. We add 20 degrees to both sides of the equation. This gives us: 5∗m(DEF^)=200o5 {*} m(\widehat{DEF}) = 200^ {o}. Now that we've isolated the term with m(DEF^)m(\widehat{DEF}), we can solve for it by dividing both sides of the equation by 5. That leaves us with: m(DEF^)=40om(\widehat{DEF}) = 40^ {o}.

So, we found that angle DEF measures 40 degrees. But wait, we're not done yet! The question asks for the measure of angle ABC, not DEF. But we're close! Now we can use the value of m(DEF^)m(\widehat{DEF}) and plug it back into either of our original equations to find m(ABC^)m(\widehat{ABC}). Let's use the second equation, m(ABC^)=4∗m(DEF^)−20om(\widehat{ABC}) = 4 {*} m(\widehat{DEF}) - 20^ {o}. We substitute m(DEF^)=40om(\widehat{DEF}) = 40^ {o} into the equation: m(ABC^)=4∗40o−20om(\widehat{ABC}) = 4 {*} 40^ {o} - 20^ {o}.

Now, perform the multiplication: m(ABC^)=160o−20om(\widehat{ABC}) = 160^ {o} - 20^ {o}. Finally, subtract to find the measure of angle ABC: m(ABC^)=140om(\widehat{ABC}) = 140^ {o}. Therefore, the measure of angle ABC is 140 degrees. Isn't it cool how we used algebra and our knowledge of supplementary angles to find the solution? This method is super important for many geometry problems. So, if you've followed along, pat yourself on the back, you've conquered a geometry problem!

Verification and Conclusion: Checking Your Work

Alright, champs, we've gone through the whole process, and we've got an answer. But before we proudly display our solution, it's always smart to check our work! That's right, we double-check our work. A quick verification step can save us from making silly mistakes. How do we verify our answer? We can use the original information to see if our solution makes sense. Remember that the original problem told us that ABC^\widehat{ABC} and DEF^\widehat{DEF} are supplementary angles, which means they must add up to 180o180^ {o}.

We found that m(ABC^)=140om(\widehat{ABC}) = 140^ {o} and m(DEF^)=40om(\widehat{DEF}) = 40^ {o}. If we add those two angles, we get 140o+40o=180o140^ {o} + 40^ {o} = 180^ {o}. That means the first condition of the problem is satisfied. And that means we didn't screw up. Now we can check the relationship between the two angles. The problem stated that the measure of angle ABC is 4 times the measure of angle DEF minus 20 degrees. So, is 140o=(4∗40o)−20o140^ {o} = (4 {*} 40^ {o}) - 20^ {o}? Let's check: 4∗40o=160o4 {*} 40^ {o} = 160^ {o} and 160o−20o=140o160^ {o} - 20^ {o} = 140^ {o}. The second condition is also satisfied, so our solution appears correct! Congratulations! You successfully solved the problem and found the measure of angle ABC. Remember, the key takeaways from this problem are:

  • Understanding Supplementary Angles: Knowing that supplementary angles add up to 180 degrees is vital. 🔑
  • Algebraic Manipulation: Using substitution and simplifying equations is a fundamental skill. 📐
  • Checking Your Work: Always verify your answer to avoid silly mistakes. ✔

Keep practicing these types of problems, and you'll become a geometry whiz in no time. If you got stuck at any point, go back and review the steps, or try similar problems to reinforce your understanding. Keep learning, keep practicing, and you'll do great. Until next time, keep exploring the fascinating world of mathematics! 🎉