Answer :
Sure, let's solve the problem step-by-step.
1. Understand the properties of a parallelogram:
- Opposite angles of a parallelogram are equal.
- The sum of the adjacent angles is 180 degrees.
2. Given angles:
- [tex]\(\angle M = 11x\)[/tex]
- [tex]\(\angle N = 6x - 7\)[/tex]
3. Using the angle sum property of adjacent angles:
- The sum of [tex]\(\angle M\)[/tex] and [tex]\(\angle N\)[/tex] should be 180 degrees:
[tex]\[ 11x + (6x - 7) = 180 \][/tex]
4. Setting up the equation:
[tex]\[ 11x + 6x - 7 = 180 \][/tex]
- Combine like terms:
[tex]\[ 17x - 7 = 180 \][/tex]
- Solve for [tex]\(x\)[/tex]:
[tex]\[ 17x = 187 \\ x = 11 \][/tex]
5. Calculate the specific 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]
6. Determine angles [tex]\(\angle L\)[/tex] and [tex]\(\angle O\)[/tex]:
- Since [tex]\(\angle M\)[/tex] and [tex]\(\angle L\)[/tex] are opposite angles of the parallelogram:
[tex]\[ m \angle L = m \angle M = 121^\circ \][/tex]
- Similarly, [tex]\(\angle N\)[/tex] and [tex]\(\angle O\)[/tex] are opposite angles:
[tex]\[ m \angle O = m \angle N = 59^\circ \][/tex]
7. Evaluating the statements:
- [tex]\(x = 11\)[/tex] is true.
- [tex]\(m \angle L = 22^\circ\)[/tex] is false because [tex]\(m \angle L = 121^\circ\)[/tex].
- [tex]\(m \angle M = 111^\circ\)[/tex] is false because [tex]\(m \angle M = 121^\circ\)[/tex].
- [tex]\(m \angle N = 59^\circ\)[/tex] is true.
- [tex]\(m \angle O = 121^\circ\)[/tex] is false because [tex]\(m \angle O = 59^\circ\)[/tex].
Thus, the true statements about the parallelogram LMNO are:
1. [tex]\(x = 11\)[/tex]
2. [tex]\(m \angle N = 59^\circ\)[/tex]
Identifying the three options to select, we get:
1. [tex]\(x = 11\)[/tex]
2. [tex]\(m \angle N = 59^\circ\)[/tex]
3. [tex]\(There is no third true statement from the provided options, only these two are correct.\)[/tex]
1. Understand the properties of a parallelogram:
- Opposite angles of a parallelogram are equal.
- The sum of the adjacent angles is 180 degrees.
2. Given angles:
- [tex]\(\angle M = 11x\)[/tex]
- [tex]\(\angle N = 6x - 7\)[/tex]
3. Using the angle sum property of adjacent angles:
- The sum of [tex]\(\angle M\)[/tex] and [tex]\(\angle N\)[/tex] should be 180 degrees:
[tex]\[ 11x + (6x - 7) = 180 \][/tex]
4. Setting up the equation:
[tex]\[ 11x + 6x - 7 = 180 \][/tex]
- Combine like terms:
[tex]\[ 17x - 7 = 180 \][/tex]
- Solve for [tex]\(x\)[/tex]:
[tex]\[ 17x = 187 \\ x = 11 \][/tex]
5. Calculate the specific 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]
6. Determine angles [tex]\(\angle L\)[/tex] and [tex]\(\angle O\)[/tex]:
- Since [tex]\(\angle M\)[/tex] and [tex]\(\angle L\)[/tex] are opposite angles of the parallelogram:
[tex]\[ m \angle L = m \angle M = 121^\circ \][/tex]
- Similarly, [tex]\(\angle N\)[/tex] and [tex]\(\angle O\)[/tex] are opposite angles:
[tex]\[ m \angle O = m \angle N = 59^\circ \][/tex]
7. Evaluating the statements:
- [tex]\(x = 11\)[/tex] is true.
- [tex]\(m \angle L = 22^\circ\)[/tex] is false because [tex]\(m \angle L = 121^\circ\)[/tex].
- [tex]\(m \angle M = 111^\circ\)[/tex] is false because [tex]\(m \angle M = 121^\circ\)[/tex].
- [tex]\(m \angle N = 59^\circ\)[/tex] is true.
- [tex]\(m \angle O = 121^\circ\)[/tex] is false because [tex]\(m \angle O = 59^\circ\)[/tex].
Thus, the true statements about the parallelogram LMNO are:
1. [tex]\(x = 11\)[/tex]
2. [tex]\(m \angle N = 59^\circ\)[/tex]
Identifying the three options to select, we get:
1. [tex]\(x = 11\)[/tex]
2. [tex]\(m \angle N = 59^\circ\)[/tex]
3. [tex]\(There is no third true statement from the provided options, only these two are correct.\)[/tex]
