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.
$x=11$
$m \angle L =22^{\circ}$
$m \angle M =111^{\circ}$
$m \angle N=59^{\circ}$
$m \angle O =121^{\circ}$
Answer (2)
Find the value of x using the property that consecutive angles in a parallelogram are supplementary: 11 x + 6 x − 7 = 180 , which simplifies to x = 11 .
Determine the measures of angles M and N by substituting the value of x: ∠ M = 11 ( 11 ) = 12 1 ∘ and ∠ N = 6 ( 11 ) − 7 = 5 9 ∘ .
Use the property that opposite angles in a parallelogram are equal to find the measures of angles L and O: ∠ L = ∠ N = 5 9 ∘ and ∠ O = ∠ M = 12 1 ∘ .
Identify the true statements based on the calculated values: 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 ) ∘ . In a parallelogram, consecutive angles are supplementary, meaning they add up to 18 0 ∘ . Also, opposite angles are equal.
Solve for x Since ∠ M and ∠ N are consecutive angles in parallelogram LMNO, we can write the equation: ∠ M + ∠ N = 18 0 ∘ Substituting the given expressions: ( 11 x ) + ( 6 x − 7 ) = 180 Combining like terms: 17 x − 7 = 180 Adding 7 to both sides: 17 x = 187 Dividing by 17: x = 17 187 = 11
Find angle M and angle N 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 angle L and angle O In a parallelogram, opposite angles are equal. Therefore, ∠ L = ∠ N and ∠ O = ∠ M . So, ∠ L = 5 9 ∘ ∠ O = 12 1 ∘
Check the statements Now, let's check the given statements:
x = 11 (True)
m ∠ L = 2 2 ∘ (False, ∠ L = 5 9 ∘ )
m ∠ M = 11 1 ∘ (False, ∠ M = 12 1 ∘ )
m ∠ N = 5 9 ∘ (True)
m ∠ O = 12 1 ∘ (True)
Identify true statements The true statements are:
x = 11
m ∠ N = 5 9 ∘
m ∠ O = 12 1 ∘
Examples
Parallelograms are commonly found in architecture and design. For example, the support structures of some bridges and buildings use parallelogram shapes for stability. Knowing the angle measures in a parallelogram helps engineers ensure that the structures are properly aligned and balanced, which is crucial for safety and functionality. Also, understanding the properties of parallelograms is essential in computer graphics for creating and manipulating shapes in 2D and 3D environments.
In parallelogram LMNO, we find that x = 11 , m ∠ N = 5 9 ∘ , and m ∠ O = 12 1 ∘ by applying properties of parallelograms. The consecutive angles M and N are supplementary, which helped us determine the values. Therefore, the correct true statements are those relating to x, angle N, and angle O.
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