Let's start by understanding the properties of a parallelogram, particularly the relationships between its angles. In a parallelogram, opposite angles are equal, and adjacent angles are supplementary (adding up to 180°). Given:
- [tex]\(\angle M = 11x\)[/tex] degrees
- [tex]\(\angle N = 6x - 7\)[/tex] degrees
We need to find the value of [tex]\( x \)[/tex] that satisfies the conditions of the parallelogram.
Since [tex]\(\angle M\)[/tex] and [tex]\(\angle N\)[/tex] are adjacent angles, they must be supplementary. Hence, we can set up the following equation:
[tex]\[ 11x + (6x - 7) = 180 \][/tex]
Let's solve this equation for [tex]\( x \)[/tex]:
[tex]\[ 11x + 6x - 7 = 180 \][/tex]
[tex]\[ 17x - 7 = 180 \][/tex]
[tex]\[ 17x = 187 \][/tex]
[tex]\[ x = 11 \][/tex]
Now, using [tex]\( x = 11 \)[/tex], we can find the measures of the specified angles:
1. [tex]\(\angle M = 11x = 11 \times 11 = 121^\circ\)[/tex]
2. [tex]\(\angle N = 6x - 7 = 6 \times 11 - 7 = 66 - 7 = 59^\circ\)[/tex]
Since LMNO is a parallelogram, [tex]\(\angle L = \angle N\)[/tex] and [tex]\(\angle O = \angle M\)[/tex]:
3. [tex]\(\angle L = \angle N = 59^\circ\)[/tex]
4. [tex]\(\angle O = \angle M = 121^\circ\)[/tex]
Given these results, let's analyze the statements provided:
1. [tex]\( x = 11 \)[/tex]: This statement is true.
2. [tex]\( m\angle L = 22^\circ \)[/tex]: This statement is false because [tex]\(\angle L = 59^\circ\)[/tex].
3. [tex]\( m\angle M = 111^\circ \)[/tex]: This statement is false because [tex]\(\angle M = 121^\circ\)[/tex].
4. [tex]\( m\angle N = 59^\circ \)[/tex]: This statement is true.
5. [tex]\( m\angle O = 121^\circ \)[/tex]: This statement is true.
In conclusion, the three true statements about parallelogram LMNO are:
- [tex]\( x = 11 \)[/tex]
- [tex]\( m\angle N = 59^\circ \)[/tex]
- [tex]\( m\angle O = 121^\circ \)[/tex]
