Let's go through the information given in the problem step-by-step to determine the values of [tex]\( x \)[/tex] and the measures of the angles in parallelogram LMNO.
1. Define the angles in terms of [tex]\( x \)[/tex]:
- [tex]\(\angle M = (11x)^\circ\)[/tex]
- [tex]\(\angle N = (6x - 7)^\circ\)[/tex]
2. Use the property of a parallelogram:
The sum of the measures of the adjacent angles in a parallelogram is [tex]\(180^\circ\)[/tex]. Therefore:
[tex]\[
\angle M + \angle N = 180^\circ
\][/tex]
Plugging in the expressions for [tex]\(\angle M\)[/tex] and [tex]\(\angle N\)[/tex]:
[tex]\[
11x + (6x - 7) = 180
\][/tex]
3. Solve for [tex]\( x \)[/tex]:
Simplify the equation:
[tex]\[
11x + 6x - 7 = 180
\][/tex]
Combine like terms:
[tex]\[
17x - 7 = 180
\][/tex]
Isolate [tex]\( x \)[/tex]:
[tex]\[
17x = 187
\][/tex]
Divide both sides by 17:
[tex]\[
x = 11
\][/tex]
4. Calculate the measures of the angles using [tex]\( x = 11 \)[/tex]:
- [tex]\(\angle M = 11x = 11 \times 11 = 121^\circ\)[/tex]
- [tex]\(\angle N = 6x - 7 = 6 \times 11 - 7 = 66 - 7 = 59^\circ\)[/tex]
- [tex]\(\angle L\)[/tex] and [tex]\(\angle M\)[/tex] are opposite angles. Therefore, [tex]\(\angle L = \angle M = 121^\circ\)[/tex].
- [tex]\(\angle O\)[/tex] is supplementary to [tex]\(\angle N\)[/tex]. Since adjacent angles in a parallelogram sum to [tex]\(180^\circ \)[/tex]:
[tex]\[
\angle O = 180^\circ - \angle N = 180 - 59 = 121^\circ
\][/tex]
Based on these calculations:
- The value of [tex]\( x \)[/tex] is [tex]\(11\)[/tex].
- The measure of [tex]\(\angle M\)[/tex] is [tex]\(121^\circ\)[/tex].
- The measure of [tex]\(\angle N\)[/tex] is [tex]\(59^\circ\)[/tex].
- The measure of [tex]\(\angle O\)[/tex] is [tex]\(121^\circ\)[/tex].
Therefore, the true statements about parallelogram LMNO are:
- [tex]\( x = 11 \)[/tex]
- [tex]\( m \angle N = 59^\circ \)[/tex]
- [tex]\( m \angle O = 121^\circ \)[/tex]
Hence, the correct options are:
- [tex]\( x = 11 \)[/tex]
- [tex]\( m \angle N = 59^\circ \)[/tex]
- [tex]\( m \angle O = 121^\circ \)[/tex]
