Certainly! Let's go through the solution step-by-step to determine the values of the angles and select the true statements about the parallelogram LMNO.
1. We start with the given angles:
- [tex]\( \angle M = 11x \)[/tex]
- [tex]\( \angle N = 6x - 7 \)[/tex]
2. In a parallelogram, opposite angles are equal. Thus:
- [tex]\( \angle M = \angle O \)[/tex]
- [tex]\( \angle N = \angle L \)[/tex]
3. The sum of the internal angles in any quadrilateral is 360 degrees. For a parallelogram:
[tex]\[
\angle M + \angle N + \angle O + \angle L = 360^\circ
\][/tex]
Since [tex]\( \angle M = \angle O \)[/tex] and [tex]\( \angle N = \angle L \)[/tex], we can simplify this to:
[tex]\[
2\angle M + 2\angle N = 360^\circ
\][/tex]
Dividing both sides by 2:
[tex]\[
\angle M + \angle N = 180^\circ
\][/tex]
4. Substitute the expressions for [tex]\( \angle M \)[/tex] and [tex]\( \angle N \)[/tex] into the equation:
[tex]\[
11x + (6x - 7) = 180
\][/tex]
Combine like terms:
[tex]\[
17x - 7 = 180
\][/tex]
Add 7 to both sides:
[tex]\[
17x = 187
\][/tex]
Divide by 17:
[tex]\[
x = 11
\][/tex]
5. Now, substitute [tex]\( x = 11 \)[/tex] back into the expressions for the angles:
- [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]\( \angle L = \angle N \)[/tex] and [tex]\( \angle O = \angle M \)[/tex]:
- [tex]\( \angle L = 59^\circ \)[/tex]
- [tex]\( \angle O = 121^\circ \)[/tex]
6. Now we can determine which statements are true:
- [tex]\( x = 11 \)[/tex] is true.
- [tex]\( m \angle L = 22^\circ \)[/tex] is false (since [tex]\( \angle L = 59^\circ \)[/tex]).
- [tex]\( m_{\angle} M = 111^\circ \)[/tex] is false (since [tex]\( \angle M = 121^\circ \)[/tex]).
- [tex]\( m \angle N = 59^\circ \)[/tex] is true.
- [tex]\( m_{\angle} O = 121^\circ \)[/tex] is true.
So the true statements about the parallelogram LMNO are:
- [tex]\( x = 11 \)[/tex]
- [tex]\( m \angle N = 59^\circ \)[/tex]
- [tex]\( m_{\angle} O = 121^\circ \)[/tex]
