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)
The true statements about parallelogram LMNO are: x = 11 , m ∠ N = 5 9 ∘ , and m ∠ O = 12 1 ∘ . By using the properties of supplementary angles in parallelograms, the measures of angles were calculated. These findings confirm the behavior of angles in parallelograms, especially regarding their relationships and equalities.
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Solve for x using the supplementary angle property: 11 x + 6 x − 7 = 180 , which simplifies to x = 11 .
Calculate ∠ M and ∠ N using x = 11 : ∠ M = 11 × 11 = 12 1 ∘ and ∠ N = 6 × 11 − 7 = 5 9 ∘ .
Determine ∠ L and ∠ O using the property of opposite angles: ∠ L = ∠ N = 5 9 ∘ and ∠ O = ∠ M = 12 1 ∘ .
Identify the true statements: x = 11 , m ∠ N = 5 9 ∘ , and m ∠ O = 12 1 ∘ .
x = 11 , m ∠ N = 5 9 ∘ , m ∠ O = 12 1 ∘
Explanation
Problem Analysis 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 their sum is 18 0 ∘ . We will use this property to find the value of x and then determine the measures of all angles in the parallelogram.
Solve for x Since ∠ M and ∠ N are consecutive angles in parallelogram LMNO, we have: ∠ 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
Calculate the angles Now that we have found x = 11 , 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 ∘ In a parallelogram, opposite angles are equal. Therefore, ∠ O = ∠ M and ∠ L = ∠ N . So, ∠ O = 12 1 ∘ ∠ L = 5 9 ∘
Determine the true statements Now, let's check which of the given statements are true:
x = 11 - This is true based on our calculation.
m ∠ L = 2 2 ∘ - This is false. We found that m ∠ L = 5 9 ∘ .
m ∠ M = 11 1 ∘ - This is false. We found that m ∠ M = 12 1 ∘ .
m ∠ N = 5 9 ∘ - This is true based on our calculation.
m ∠ O = 12 1 ∘ - This is true based on our calculation.
Therefore, the true statements are:
x = 11
m ∠ N = 5 9 ∘
m ∠ O = 12 1 ∘
Examples
Understanding the properties of parallelograms is useful in architecture and construction. For example, when designing buildings or structures with parallel sides, knowing that opposite angles are equal and consecutive angles are supplementary helps ensure structural integrity and aesthetic balance. This knowledge is also applicable in creating accurate blueprints and models.
