To determine the correct statements about the parallelogram LMNO, let's go through a detailed, step-by-step solution.
### Step-by-Step Solution:
1. Given:
- ∠M = [tex]\( (11x)^\circ \)[/tex]
- ∠N = [tex]\( (6x - 7)^\circ \)[/tex]
2. Property of Parallelograms:
- Opposite angles are equal.
- Consecutive angles are supplementary, meaning they add up to 180°.
3. Set Up the Equation:
Since ∠M and ∠N are consecutive angles:
[tex]\[
(11x) + (6x - 7) = 180^\circ
\][/tex]
4. Combine Like Terms:
[tex]\[
11x + 6x - 7 = 180
\][/tex]
[tex]\[
17x - 7 = 180
\][/tex]
5. Solve for x:
[tex]\[
17x = 187
\][/tex]
[tex]\[
x = \frac{187}{17} = 11
\][/tex]
6. Calculate the Angles Using the Value of x:
- ∠M:
[tex]\[
(11x)^\circ = 11 \times 11 = 121^\circ
\][/tex]
- ∠N:
[tex]\[
(6x - 7)^\circ = 6 \times 11 - 7 = 66 - 7 = 59^\circ
\][/tex]
- ∠L and ∠O:
Since opposite angles in a parallelogram are equal:
[tex]\[
\angle L = \angle M = 121^\circ
\][/tex]
[tex]\[
\angle O = \angle N = 59^\circ
\][/tex]
### Summary and Statements Verification:
1. x = 11
- This statement is true.
2. [tex]\( m\angle L = 22^\circ \)[/tex]
- This statement is false. We found that [tex]\( m\angle L = 121^\circ \)[/tex].
3. [tex]\( m\angle M = 111^\circ \)[/tex]
- This statement is false. We found that [tex]\( m\angle M = 121^\circ \)[/tex].
4. [tex]\( m\angle N = 59^\circ \)[/tex]
- This statement is true. We calculated [tex]\( m\angle N \)[/tex] to be 59°.
5. [tex]\( m\angle O = 121^\circ \)[/tex]
- This statement is true. We found that [tex]\( m\angle O = 121^\circ \)[/tex].
### Correct Statements:
- [tex]\( x = 11 \)[/tex]
- [tex]\( m\angle N = 59^\circ \)[/tex]
- [tex]\( m\angle O = 121^\circ \)[/tex]
These are the three correct statements about parallelogram LMNO.
