To determine which statements are true about the parallelogram LMNO, let's analyze the given information step-by-step.
1. Understanding the Problem:
- We are given a parallelogram LMNO.
- [tex]\(\angle M = (11x)^\circ\)[/tex]
- [tex]\(\angle N = (6x - 7)^\circ\)[/tex]
2. Properties of a Parallelogram:
- Opposite angles in a parallelogram are equal.
- Adjacent angles in a parallelogram are supplementary (sum to [tex]\(180^\circ\)[/tex]).
3. Set Up the Equation:
Since [tex]\(\angle M\)[/tex] and [tex]\(\angle N\)[/tex] are on the same side in a parallelogram, they are supplementary:
[tex]\[
\angle M + \angle N = 180^\circ
\][/tex]
Substituting the given expressions:
[tex]\[
(11x) + (6x - 7) = 180
\][/tex]
4. Solve for [tex]\(x\)[/tex]:
Combine like terms:
[tex]\[
11x + 6x - 7 = 180
\][/tex]
[tex]\[
17x - 7 = 180
\][/tex]
Add 7 to both sides:
[tex]\[
17x = 187
\][/tex]
Divide both sides by 17:
[tex]\[
x = 11
\][/tex]
5. Determine the Measure of Each Angle:
- [tex]\(\angle M = 11x = 11 \times 11 = 121^\circ\)[/tex]
- [tex]\(\angle N = 6x - 7 = 6 \times 11 - 7 = 66 - 7 = 59^\circ\)[/tex]
- Since [tex]\(\angle L\)[/tex] is opposite to [tex]\(\angle N\)[/tex], [tex]\(\angle L = 59^\circ\)[/tex]
- [tex]\(\angle O\)[/tex] is opposite to [tex]\(\angle M\)[/tex], [tex]\(\angle O = 121^\circ\)[/tex]
6. Verify the Statements:
- [tex]\(x = 11\)[/tex]: True
- [tex]\(m \angle L = 22^\circ\)[/tex]: False (should be [tex]\(59^\circ\)[/tex])
- [tex]\(m_{\angle}M = 111^\circ\)[/tex]: False (should be [tex]\(121^\circ\)[/tex])
- [tex]\(m \angle N = 59^\circ\)[/tex]: True
- [tex]\(m_{\angle}O = 121^\circ\)[/tex]: True
Therefore, the three true statements about parallelogram LMNO are:
1. [tex]\(x = 11\)[/tex]
2. [tex]\(m \angle N = 59^\circ\)[/tex]
3. [tex]\(m_{\angle}O = 121^\circ\)[/tex]
