Question

lmno is a parallelogram, with \\(\angle m = (11x)^\circ\\) and \\(\angle n = (6x - 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\\)

Explanation:

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<reasoning>

Set up the angle relationship

In a parallelogram \(LMNO\), consecutive angles are supplementary. Since \(M\) and \(N\) are consecutive vertices in the parallelogram \(LMNO\), their angles must add up to \(180^\circ\):
\[m\angle M + m\angle N = 180^\circ\]

Solve for x

Using the Solving Linear Equations knowledge point
\[

$$\begin{aligned} (11x) + (6x - 7) &= 180 \\ 17x - 7 &= 180 \\ 17x &= 187 \\ x &= 11 \end{aligned}$$

\]

Calculate the angle measures

Now we substitute \(x = 11\) back into the expressions for each angle:

• \(m\angle M = 11(11) = 121^\circ\)
• \(m\angle N = 6(11) - 7 = 66 - 7 = 59^\circ\)

In a parallelogram, opposite angles are equal:

• \(m\angle L = m\angle N = 59^\circ\)
• \(m\angle O = m\angle M = 121^\circ\)

Evaluate the given statements

Let's check each option against our calculated values:

• \(x = 11\): True.
• \(m\angle L = 22^\circ\): False (it is \(59^\circ\)).
• \(m\angle M = 111^\circ\): False (it is \(121^\circ\)).
• \(m\angle N = 59^\circ\): True.
• \(m\angle O = 121^\circ\): True.

</reasoning>

<answer>
<mcq-correct>\(x = 11\)</mcq-correct>
<mcq-option>\(m\angle L = 22^\circ\)</mcq-option>
<mcq-option>\(m\angle M = 111^\circ\)</mcq-option>
<mcq-correct>\(m\angle N = 59^\circ\)</mcq-correct>
<mcq-correct>\(m\angle O = 121^\circ\)</mcq-correct>
</answer>

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Answer:

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<reasoning>

Set up the angle relationship

In a parallelogram \(LMNO\), consecutive angles are supplementary. Since \(M\) and \(N\) are consecutive vertices in the parallelogram \(LMNO\), their angles must add up to \(180^\circ\):
\[m\angle M + m\angle N = 180^\circ\]

Solve for x

Using the Solving Linear Equations knowledge point
\[

$$\begin{aligned} (11x) + (6x - 7) &= 180 \\ 17x - 7 &= 180 \\ 17x &= 187 \\ x &= 11 \end{aligned}$$

\]

Calculate the angle measures

Now we substitute \(x = 11\) back into the expressions for each angle:

• \(m\angle M = 11(11) = 121^\circ\)
• \(m\angle N = 6(11) - 7 = 66 - 7 = 59^\circ\)

In a parallelogram, opposite angles are equal:

• \(m\angle L = m\angle N = 59^\circ\)
• \(m\angle O = m\angle M = 121^\circ\)

Evaluate the given statements

Let's check each option against our calculated values:

• \(x = 11\): True.
• \(m\angle L = 22^\circ\): False (it is \(59^\circ\)).
• \(m\angle M = 111^\circ\): False (it is \(121^\circ\)).
• \(m\angle N = 59^\circ\): True.
• \(m\angle O = 121^\circ\): True.

</reasoning>

<answer>
<mcq-correct>\(x = 11\)</mcq-correct>
<mcq-option>\(m\angle L = 22^\circ\)</mcq-option>
<mcq-option>\(m\angle M = 111^\circ\)</mcq-option>
<mcq-correct>\(m\angle N = 59^\circ\)</mcq-correct>
<mcq-correct>\(m\angle O = 121^\circ\)</mcq-correct>
</answer>

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