Decoding Parallelogram LMNO A Step-by-Step Solution

by ADMIN 52 views

Hey guys! Today, we're diving into the fascinating world of parallelograms, specifically parallelogram LMNO. We've got some angle measurements to work with, and our mission is to figure out which statements are true about this four-sided shape. So, let's put on our geometry hats and get started!

Understanding the Problem: Parallelogram LMNO and Its Angles

In this geometrical puzzle, we are given a parallelogram named LMNO. Remember, a parallelogram is a quadrilateral (a four-sided shape) with opposite sides that are parallel and equal in length. A crucial property of parallelograms that we'll use here is that opposite angles are equal, and adjacent angles are supplementary (meaning they add up to 180 degrees). We're given that $\angle M = (11x)^{\circ}$ and $\angle N = (6x - 7)^{\circ}$. Our task is to determine which statements about this parallelogram are true. We have three options to choose from, which likely involve the value of x and the measures of some angles.

To tackle this, our initial focus needs to be on finding the value of x. Since angles M and N are adjacent angles in the parallelogram, we know that they are supplementary. This is our golden ticket to solving for x. Once we find x, we can then calculate the actual angle measures and evaluate the given statements.

Think of it like this: we're detectives, and x is our first clue. Once we crack the x code, we can unlock the mysteries of the parallelogram's angles. We'll use the properties of parallelograms as our magnifying glass, carefully examining the relationships between the angles to reveal the truth. It's like a mathematical treasure hunt, and the treasure is the correct statements about parallelogram LMNO!

Remember, the key to success here is understanding the properties of parallelograms. Opposite angles are equal, adjacent angles are supplementary, and opposite sides are parallel and equal. These are the tools in our geometry toolkit, and we'll use them to dissect this problem and arrive at the correct solution. So, let's roll up our sleeves and get to work on unraveling the secrets of parallelogram LMNO.

Solving for x: The Key to Unlocking the Angles

Our first major goal is to determine the value of x. As we discussed, angles M and N are adjacent angles in parallelogram LMNO. This means they are supplementary, and their measures add up to 180 degrees. We can write this relationship as an equation:

∠M+∠N=180∘\angle M + \angle N = 180^{\circ}

Now, let's substitute the given expressions for the angle measures:

(11x)∘+(6x−7)∘=180∘(11x)^{\circ} + (6x - 7)^{\circ} = 180^{\circ}

This equation is our roadmap to finding x. It's a simple algebraic equation that we can solve by combining like terms and isolating x. First, let's combine the x terms:

11x+6x−7=18011x + 6x - 7 = 180

17x−7=18017x - 7 = 180

Next, we'll add 7 to both sides of the equation to isolate the term with x:

17x−7+7=180+717x - 7 + 7 = 180 + 7

17x=18717x = 187

Finally, to solve for x, we'll divide both sides of the equation by 17:

17x17=18717\frac{17x}{17} = \frac{187}{17}

x=11x = 11

Boom! We've found the value of x. It turns out that x equals 11. This is a crucial piece of information because now we can use it to calculate the measures of angles M and N. This is like finding the key that unlocks the rest of the puzzle. With x in hand, we're ready to move on to the next step: determining the actual angle measures in parallelogram LMNO.

Finding x is often the critical first step in geometry problems involving angles and algebraic expressions. It's like setting the foundation for a building; once you have a solid foundation, you can build upon it to reach the final solution. In this case, x = 11 is our foundation, and we'll use it to construct our understanding of the angles in parallelogram LMNO.

Calculating the Angle Measures: Putting x to Work

Now that we know x = 11, we can calculate the measures of angles M and N. Remember, we were given that $\angle M = (11x)^{\circ}$ and $\angle N = (6x - 7)^{\circ}$. Let's plug in x = 11 into these expressions:

For angle M:

∠M=(11∗11)∘=121∘\angle M = (11 * 11)^{\circ} = 121^{\circ}

So, angle M measures 121 degrees.

For angle N:

∠N=(6∗11−7)∘=(66−7)∘=59∘\angle N = (6 * 11 - 7)^{\circ} = (66 - 7)^{\circ} = 59^{\circ}

Therefore, angle N measures 59 degrees.

Now we have the measures of two angles in the parallelogram. But remember, a parallelogram has four angles, and we need to figure out the measures of angles L and O as well. This is where the properties of parallelograms come in handy again. We know that opposite angles in a parallelogram are equal. This means that angle L is opposite angle N, and angle O is opposite angle M.

Therefore:

∠L=∠N=59∘\angle L = \angle N = 59^{\circ}

∠O=∠M=121∘\angle O = \angle M = 121^{\circ}

We've now successfully calculated the measures of all four angles in parallelogram LMNO: $\angle M = 121^{\circ}$, $\angle N = 59^{\circ}$, $\angle L = 59^{\circ}$, and $\angle O = 121^{\circ}$. This is a significant achievement because with these angle measures, we can now evaluate the statements given in the problem and determine which ones are true.

Calculating the angle measures is like filling in the missing pieces of a puzzle. We started with a value for x, used it to find two angle measures, and then leveraged the properties of parallelograms to deduce the remaining angles. This process highlights the interconnectedness of geometry; knowing one piece of information can often lead you to discovering others. Now that we have the full picture of the angles in parallelogram LMNO, we're ready to tackle the final step: verifying the given statements.

Verifying the Statements: Unveiling the Truth about Parallelogram LMNO

With the value of x and the measures of all the angles in hand, we're now ready to evaluate the statements provided in the problem. This is where we put all our hard work to the test and see which statements hold true for parallelogram LMNO.

Let's revisit the statements:

  1. x = 11

  2. m∠L=22∘m \angle L = 22^{\circ}

  3. m∠O=121∘m \angle O = 121^{\circ}

We'll examine each statement one by one, comparing them to our calculated values.

Statement 1: x = 11

We found that x = 11 when we solved the equation $17x = 187$. So, this statement is TRUE.

Statement 2: $m \angle L = 22^{\circ}$

We calculated that $\angle L = 59^{\circ}$. This statement claims that $\angle L = 22^{\circ}$, which is incorrect. Therefore, this statement is FALSE.

Statement 3: $m \angle O = 121^{\circ}$

We found that $\angle O = 121^{\circ}$. This statement matches our calculated value, so this statement is TRUE.

So, after careful verification, we've determined that the true statements about parallelogram LMNO are x = 11 and $m \angle O = 121^{\circ}$. This is the culmination of our geometrical journey, where we started with angle expressions, solved for x, calculated angle measures, and finally, verified the given statements.

Verifying statements is a crucial step in problem-solving. It's like the quality control check in a manufacturing process, ensuring that the final product meets the required specifications. In this case, the "product" is our solution, and we've verified that it aligns with the properties of parallelograms and the given information. We can confidently say that we've successfully decoded parallelogram LMNO and identified the true statements about it. Remember, the correct options to select are:

  • x=11

  • m∠O=121∘m \angle O = 121^{\circ}

Conclusion: Mastering Parallelograms and Geometry Problems

Great job, everyone! We've successfully navigated the world of parallelograms and solved a challenging geometry problem. We started with a parallelogram, angle expressions, and a mission to uncover the truth. Through careful application of parallelogram properties, algebraic manipulation, and logical deduction, we found the value of x, calculated angle measures, and verified statements. This journey highlights the power of geometry and the importance of understanding shapes and their properties.

This problem wasn't just about finding the right answers; it was about developing our problem-solving skills. We learned to break down a complex problem into smaller, manageable steps. We applied the properties of parallelograms, solved algebraic equations, and verified our results. These are valuable skills that can be applied to a wide range of problems, both in mathematics and in life.

Remember, the key to mastering geometry is practice and a deep understanding of fundamental concepts. Don't be afraid to tackle challenging problems; they're opportunities to grow and strengthen your skills. Keep exploring, keep questioning, and keep solving! Geometry is a fascinating world filled with shapes, patterns, and relationships, and there's always something new to discover.

So, the next time you encounter a parallelogram, you'll be well-equipped to tackle it. You'll know how to use angle relationships, solve for variables, and apply the properties of parallelograms to uncover hidden truths. Keep up the great work, and happy problem-solving!