Let's solve the problem step by step.
1. Identify the property of parallelograms regarding angles:
- In any parallelogram, the sum of the measures of two adjacent angles is [tex]\(180^\circ\)[/tex].
2. Set up the equation using the given angles:
- We're given that [tex]\(\angle M = 11x^\circ\)[/tex] and [tex]\(\angle N = (6x - 7)^\circ\)[/tex].
- Therefore, the sum of [tex]\(\angle M\)[/tex] and [tex]\(\angle N\)[/tex] will be:
[tex]\[
\angle M + \angle N = 180^\circ
\][/tex]
Substitute the given angle expressions:
[tex]\[
11x + (6x - 7) = 180
\][/tex]
3. Solve for [tex]\(x\)[/tex]:
- Simplify the equation:
[tex]\[
11x + 6x - 7 = 180
\][/tex]
[tex]\[
17x - 7 = 180
\][/tex]
[tex]\[
17x = 187
\][/tex]
[tex]\[
x = 11
\][/tex]
4. Verify the measures of the angles using [tex]\(x = 11\)[/tex]:
- [tex]\(\angle M = 11x = 11 \cdot 11 = 121^\circ\)[/tex]
- [tex]\(\angle N = 6x - 7 = 6 \cdot 11 - 7 = 66 - 7 = 59^\circ\)[/tex]
5. Determine the other angles in the parallelogram:
- In a parallelogram, opposite angles are equal. So, [tex]\(\angle L = \angle M\)[/tex] and [tex]\(\angle O = \angle N\)[/tex].
- Therefore, [tex]\(\angle L = 121^\circ\)[/tex] and [tex]\(\angle O = 59^\circ\)[/tex].
6. Evaluate the given statements:
- [tex]\(x = 11\)[/tex]: This is true.
- [tex]\(m \angle L = 22^\circ\)[/tex]: This is false. We found [tex]\(\angle L = 121^\circ\)[/tex].
- [tex]\(m \angle M = 111^\circ\)[/tex]: This is false. We found [tex]\(\angle M = 121^\circ\)[/tex].
- [tex]\(m \angle N = 59^\circ\)[/tex]: This is true.
- [tex]\(m \angle O = 121^\circ\)[/tex]: This is false. We found [tex]\(\angle O = 59^\circ\)[/tex].
So, the three statements that are true are:
- [tex]\(x = 11\)[/tex]
- [tex]\(m \angle N = 59^\circ\)[/tex]
These are the true statements based on the properties and given information about the parallelogram.
