LMNO is a parallelogram, with $\angle M =(11 x)^{\circ}$ and $\angle N =(6 x-7)^{\circ}$. Which statements are true about parallelogram LMNO? Select three options.
A. $x=11$
B. $m \angle L=22^{\circ}$
C. $m \angle M =111^{\circ}$
D. $m \angle N=59^{\circ}$
E. $m _{\angle} O =121^{\circ}$
Answer (2)
The true statements about parallelogram LMNO are that x = 11 , m ∠ N = 5 9 ∘ , and m ∠ O = 12 1 ∘ .
;
Use the property that consecutive angles in a parallelogram are supplementary: ∠ M + ∠ N = 18 0 ∘ .
Substitute the given expressions and solve for x : 11 x + 6 x − 7 = 180 ⇒ x = 11 .
Calculate the measures of the angles: ∠ M = 12 1 ∘ , ∠ N = 5 9 ∘ , ∠ L = 5 9 ∘ , ∠ O = 12 1 ∘ .
Identify the true statements based on the calculated values: x = 11 , m ∠ N = 5 9 ∘ , m ∠ O = 12 1 ∘ .
x = 11 , m ∠ N = 5 9 ∘ , m ∠ O = 12 1 ∘
Explanation
Analyze the problem Let's analyze the given information. We have a parallelogram LMNO, where ∠ M = ( 11 x ) ∘ and ∠ N = ( 6 x − 7 ) ∘ . We need to determine which of the given statements are true.
Use properties of parallelograms In a parallelogram, consecutive angles are supplementary, meaning their sum is 18 0 ∘ . Therefore, we have: ∠ M + ∠ N = 18 0 ∘ Substituting the given expressions, we get: ( 11 x ) + ( 6 x − 7 ) = 180
Solve for x Now, let's solve for x :
11 x + 6 x − 7 = 180 17 x − 7 = 180 17 x = 187 x = 17 187 x = 11
Calculate angle measures Now that we have the value of x , we can find the measures of ∠ M and ∠ N :
∠ M = 11 x = 11 ( 11 ) = 12 1 ∘ ∠ N = 6 x − 7 = 6 ( 11 ) − 7 = 66 − 7 = 5 9 ∘
Find opposite angles In a parallelogram, opposite angles are equal. Therefore: ∠ L = ∠ N = 5 9 ∘ ∠ O = ∠ M = 12 1 ∘
Verify the statements Now, let's check the given statements:
x = 11 (True)
m ∠ L = 2 2 ∘ (False, it's 5 9 ∘ )
m ∠ M = 11 1 ∘ (False, it's 12 1 ∘ )
m ∠ N = 5 9 ∘ (True)
m ∠ O = 12 1 ∘ (True)
Final Answer The true statements are:
x = 11
m ∠ N = 5 9 ∘
m ∠ O = 12 1 ∘
Examples
Parallelograms are commonly found in architecture and design, such as in the construction of bridges or the layout of rooms. Understanding the properties of parallelograms, like the relationships between their angles, is crucial for ensuring structural stability and aesthetic appeal. For example, if you're designing a room with a parallelogram-shaped window, knowing that opposite angles are equal helps you position the window correctly for optimal light and symmetry. Similarly, knowing that consecutive angles are supplementary ensures that the window fits properly within the structure of the wall, maintaining the overall integrity of the building's design.
