Solving For X In Parallelogram PQRS A Step-by-Step Guide

Hey guys! Today, we're diving into a fun geometry problem involving parallelograms. We're given a parallelogram PQRS, where the measure of angle P (m∠P) is (3x + 9) degrees, and the measure of its opposite angle R (m∠R) is (5x - 15) degrees. Our mission, should we choose to accept it, is to solve for x. Don't worry, it's not as daunting as it sounds! We'll break it down step by step, making sure everyone's on board. So, let's put on our thinking caps and get started!

Understanding Parallelograms

Before we jump into the algebra, let's quickly recap what makes a parallelogram a parallelogram. This will give us the foundation we need to tackle the problem effectively. Think of it as our parallelogram 101 crash course!

• Opposite sides are parallel: This is the defining characteristic of a parallelogram. Imagine two pairs of train tracks running side by side – that's the essence of parallel sides. In our parallelogram PQRS, side PQ is parallel to side RS, and side QR is parallel to side SP.
• Opposite sides are congruent: Not only are the opposite sides parallel, but they're also equal in length. So, PQ = RS and QR = SP. This is a neat little property that often comes in handy.
• Opposite angles are congruent: This is the key property we'll be using to solve for x. Opposite angles in a parallelogram are equal in measure. In our case, ∠P and ∠R are opposite angles, and ∠Q and ∠S are opposite angles. This means m∠P = m∠R, which is the golden nugget we need!
• Consecutive angles are supplementary: Consecutive angles are angles that are next to each other, like ∠P and ∠Q, or ∠Q and ∠R. Supplementary means they add up to 180 degrees. This property is useful in other parallelogram problems, but not directly needed for this one.

With these properties in our mental toolkit, we're well-equipped to tackle the problem. Remember, the most important property for us right now is that opposite angles in a parallelogram are congruent. Let's keep that in mind as we move forward.

Setting Up the Equation

Okay, now for the fun part – translating our geometric knowledge into an algebraic equation! This is where we take the information we have and turn it into something we can actually solve. Remember, we know that opposite angles in a parallelogram are equal. In our parallelogram PQRS, ∠P and ∠R are opposite angles. We're given that m∠P = (3x + 9)° and m∠R = (5x - 15)°. Since these angles are equal, we can set their expressions equal to each other. This gives us the equation:

(3x + 9) = (5x - 15)

This equation is the heart of our problem. It's a simple algebraic equation, but it represents the geometric relationship between the angles in our parallelogram. Once we solve for x, we'll have our answer! But before we jump into solving, let's take a moment to appreciate what we've done. We've successfully translated a geometric problem into an algebraic one, which is a crucial skill in mathematics. Now, let's roll up our sleeves and solve for x!

Solving for x: Step-by-Step

Alright, let's get down to business and solve for x. We've got our equation: (3x + 9) = (5x - 15). Now, we need to isolate x on one side of the equation. Think of it like a balancing act – whatever we do to one side, we have to do to the other to keep the equation balanced.

Here’s how we can solve it step-by-step:

1. Combine like terms: Our goal is to get all the x terms on one side and the constant terms on the other. Let's start by subtracting 3x from both sides of the equation. This will eliminate the x term on the left side:

   (3x + 9) - 3x = (5x - 15) - 3x

   This simplifies to:

   9 = 2x - 15

2. Isolate the x term: Now, we want to get the x term by itself. To do this, we need to get rid of the -15 on the right side. We can do this by adding 15 to both sides:

   9 + 15 = (2x - 15) + 15

   This simplifies to:

   24 = 2x

3. Solve for x: We're almost there! Now we have 24 = 2x. To solve for x, we need to divide both sides by 2:

   24 / 2 = (2x) / 2

   This gives us:

   12 = x

So, we've found that x = 12! Woohoo! We've successfully solved for x using our knowledge of parallelograms and some basic algebra. Now, let's take a moment to celebrate our victory and then think about what this means in the context of our original problem.

Verifying the Solution

Before we declare victory, it's always a good idea to double-check our work. This is especially important in math, where a small mistake can throw off the whole answer. Verifying our solution helps us catch any errors and ensures that we're confident in our answer. So, let's put on our detective hats and make sure our value of x = 12 works in the context of our parallelogram problem.

Remember, we were given that m∠P = (3x + 9)° and m∠R = (5x - 15)°. We found that x = 12. To verify our solution, we can substitute x = 12 into these expressions and see if the resulting angle measures are equal (since opposite angles in a parallelogram are congruent).

Let's start with m∠P:

m∠P = (3x + 9)° = (3 * 12 + 9)° = (36 + 9)° = 45°

Now, let's calculate m∠R:

m∠R = (5x - 15)° = (5 * 12 - 15)° = (60 - 15)° = 45°

Aha! We see that m∠P = 45° and m∠R = 45°. Since the measures of the opposite angles are equal, our solution x = 12 is consistent with the properties of a parallelogram. This gives us confidence that we've solved the problem correctly. Verification is a powerful tool in mathematics, and it's always worth the extra effort to ensure accuracy.

Final Answer and Implications

We've done it! We've successfully solved for x in parallelogram PQRS. We found that x = 12. Let's write that down clearly:

x = 12

This is our final answer. But what does this mean in the context of the problem? Well, we now know the value of x that makes the given angle measures consistent with the properties of a parallelogram. We even verified our solution by plugging x = 12 back into the expressions for m∠P and m∠R and confirmed that they are equal.

This problem highlights the interplay between geometry and algebra. We used our knowledge of parallelograms (specifically, the property that opposite angles are congruent) to set up an algebraic equation. Then, we used our algebraic skills to solve for x. This is a common theme in mathematics – using different branches of math to solve problems. Understanding this connection can make you a more versatile problem solver.

So, next time you see a parallelogram problem, remember the properties we discussed, and don't be afraid to translate the geometry into algebra. You've got this!

Conclusion

So, guys, we've successfully navigated the world of parallelograms and solved for x! We started by understanding the properties of parallelograms, set up an equation, solved for x step-by-step, and even verified our solution. This journey has not only given us the answer to this specific problem but has also reinforced our understanding of geometry and algebra. Remember, the key to success in math is understanding the underlying concepts and practicing problem-solving techniques. Keep exploring, keep learning, and keep having fun with math!