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Sagot :
To determine the correct statements about the parallelogram LMNO, let's start by understanding some key properties of a parallelogram and calculating the necessary angles.
1. Sum of Adjacent Angles:
In a parallelogram, the sum of any two adjacent angles is [tex]\(180^\circ\)[/tex]. Therefore, we have:
[tex]\[ \angle M + \angle N = 180^\circ \][/tex]
Given:
[tex]\[ \angle M = 11x \quad \text{and} \quad \angle N = 6x - 7 \][/tex]
Using the property of the sum of adjacent angles:
[tex]\[ 11x + (6x - 7) = 180 \][/tex]
2. Solving for [tex]\(x\)[/tex]:
Simplify the equation:
[tex]\[ 11x + 6x - 7 = 180 \][/tex]
[tex]\[ 17x - 7 = 180 \][/tex]
[tex]\[ 17x = 187 \][/tex]
[tex]\[ x = \frac{187}{17} \][/tex]
[tex]\[ x = 11 \][/tex]
3. Finding the Measure of Angles:
Using [tex]\(x = 11\)[/tex]:
- Measure of [tex]\(\angle M\)[/tex]:
[tex]\[ \angle M = 11x = 11 \cdot 11 = 121^\circ \][/tex]
- Measure of [tex]\(\angle N\)[/tex]:
[tex]\[ \angle N = 6x - 7 = 6 \cdot 11 - 7 = 66 - 7 = 59^\circ \][/tex]
- Measure of [tex]\(\angle L\)[/tex] and [tex]\(\angle O\)[/tex]:
In a parallelogram, opposite angles are equal. Therefore, [tex]\(\angle L = \angle N\)[/tex] and [tex]\(\angle O = \angle M\)[/tex].
[tex]\[ \angle L = 59^\circ \quad \text{and} \quad \angle O = 121^\circ \][/tex]
Let's summarize and check the statements provided:
1. [tex]\(x = 11\)[/tex]
- This is true, as we have calculated [tex]\(x = 11\)[/tex].
2. [tex]\(m \angle L = 22^\circ\)[/tex]
- This is false. We calculated [tex]\( \angle L = 59^\circ \)[/tex].
3. [tex]\(m \angle M = 111^\circ\)[/tex]
- This is false. We calculated [tex]\( \angle M = 121^\circ \)[/tex].
4. [tex]\(m \angle N = 59^\circ\)[/tex]
- This is true. We calculated [tex]\( \angle N = 59^\circ \)[/tex].
5. [tex]\(m \angle O = 121^\circ\)[/tex]
- This is true. We calculated [tex]\( \angle O = 121^\circ \)[/tex].
Thus, the three true statements are:
- [tex]\( x = 11 \)[/tex]
- [tex]\( m \angle N = 59^\circ \)[/tex]
- [tex]\( m \angle O = 121^\circ \)[/tex]
1. Sum of Adjacent Angles:
In a parallelogram, the sum of any two adjacent angles is [tex]\(180^\circ\)[/tex]. Therefore, we have:
[tex]\[ \angle M + \angle N = 180^\circ \][/tex]
Given:
[tex]\[ \angle M = 11x \quad \text{and} \quad \angle N = 6x - 7 \][/tex]
Using the property of the sum of adjacent angles:
[tex]\[ 11x + (6x - 7) = 180 \][/tex]
2. Solving for [tex]\(x\)[/tex]:
Simplify the equation:
[tex]\[ 11x + 6x - 7 = 180 \][/tex]
[tex]\[ 17x - 7 = 180 \][/tex]
[tex]\[ 17x = 187 \][/tex]
[tex]\[ x = \frac{187}{17} \][/tex]
[tex]\[ x = 11 \][/tex]
3. Finding the Measure of Angles:
Using [tex]\(x = 11\)[/tex]:
- Measure of [tex]\(\angle M\)[/tex]:
[tex]\[ \angle M = 11x = 11 \cdot 11 = 121^\circ \][/tex]
- Measure of [tex]\(\angle N\)[/tex]:
[tex]\[ \angle N = 6x - 7 = 6 \cdot 11 - 7 = 66 - 7 = 59^\circ \][/tex]
- Measure of [tex]\(\angle L\)[/tex] and [tex]\(\angle O\)[/tex]:
In a parallelogram, opposite angles are equal. Therefore, [tex]\(\angle L = \angle N\)[/tex] and [tex]\(\angle O = \angle M\)[/tex].
[tex]\[ \angle L = 59^\circ \quad \text{and} \quad \angle O = 121^\circ \][/tex]
Let's summarize and check the statements provided:
1. [tex]\(x = 11\)[/tex]
- This is true, as we have calculated [tex]\(x = 11\)[/tex].
2. [tex]\(m \angle L = 22^\circ\)[/tex]
- This is false. We calculated [tex]\( \angle L = 59^\circ \)[/tex].
3. [tex]\(m \angle M = 111^\circ\)[/tex]
- This is false. We calculated [tex]\( \angle M = 121^\circ \)[/tex].
4. [tex]\(m \angle N = 59^\circ\)[/tex]
- This is true. We calculated [tex]\( \angle N = 59^\circ \)[/tex].
5. [tex]\(m \angle O = 121^\circ\)[/tex]
- This is true. We calculated [tex]\( \angle O = 121^\circ \)[/tex].
Thus, the three true statements are:
- [tex]\( x = 11 \)[/tex]
- [tex]\( m \angle N = 59^\circ \)[/tex]
- [tex]\( m \angle O = 121^\circ \)[/tex]
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