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Sagot :
To determine which statements about parallelogram LMNO are true, we will need to follow a systematic approach to solve for the variable [tex]\( x \)[/tex] and calculate the angles mentioned:
1. Identify given angles:
[tex]\[ \angle M = 11x \quad \text{and} \quad \angle N = (6x - 7) \][/tex]
2. Understand the properties of a parallelogram:
- Opposite angles are equal.
- Adjacent angles sum up to [tex]\( 180^\circ \)[/tex].
3. Set up the equation for adjacent angles:
Since [tex]\( \angle M \)[/tex] and [tex]\( \angle N \)[/tex] are adjacent in a parallelogram:
[tex]\[ 11x + (6x - 7) = 180^\circ \][/tex]
4. Simplify and solve for [tex]\( x \)[/tex]:
[tex]\[ 11x + 6x - 7 = 180 \implies 17x - 7 = 180 \implies 17x = 187 \implies x = \frac{187}{17} = 11 \][/tex]
5. Calculate [tex]\( \angle M \)[/tex] (using [tex]\( x = 11 \)[/tex]):
[tex]\[ \angle M = 11x = 11 \times 11 = 121^\circ \][/tex]
6. Calculate [tex]\( \angle N \)[/tex] (using [tex]\( x = 11 \)[/tex]):
[tex]\[ \angle N = 6x - 7 = 6 \times 11 - 7 = 66 - 7 = 59^\circ \][/tex]
7. Calculate the remaining angles:
[tex]\[ \angle O = \angle M = 121^\circ \quad \text{(opposite to } \angle M \text{)} \][/tex]
[tex]\[ \angle L = \angle N = 59^\circ \quad \text{(opposite to } \angle N \text{)} \][/tex]
Now that we have all the angles, we can identify which statements are true:
1. [tex]\( x = 11 \)[/tex]:
This statement is true because we solved [tex]\( x = 11 \)[/tex].
2. [tex]\( m L L = 22^\circ \)[/tex]:
This statement is false. It does not correspond to any of the angles calculated.
3. [tex]\( m_{\angle} M = 111^\circ \)[/tex]:
This statement is false. The correct measure of [tex]\( \angle M \)[/tex] is [tex]\( 121^\circ \)[/tex].
4. [tex]\( m_{\angle} N = 59^\circ \)[/tex]:
This statement is true and matches our calculation.
5. [tex]\( m_{\angle} O = 121^\circ \)[/tex]:
This statement is true because [tex]\( \angle O \)[/tex] is opposite [tex]\( \angle M \)[/tex] and equals [tex]\( 121^\circ \)[/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. Identify given angles:
[tex]\[ \angle M = 11x \quad \text{and} \quad \angle N = (6x - 7) \][/tex]
2. Understand the properties of a parallelogram:
- Opposite angles are equal.
- Adjacent angles sum up to [tex]\( 180^\circ \)[/tex].
3. Set up the equation for adjacent angles:
Since [tex]\( \angle M \)[/tex] and [tex]\( \angle N \)[/tex] are adjacent in a parallelogram:
[tex]\[ 11x + (6x - 7) = 180^\circ \][/tex]
4. Simplify and solve for [tex]\( x \)[/tex]:
[tex]\[ 11x + 6x - 7 = 180 \implies 17x - 7 = 180 \implies 17x = 187 \implies x = \frac{187}{17} = 11 \][/tex]
5. Calculate [tex]\( \angle M \)[/tex] (using [tex]\( x = 11 \)[/tex]):
[tex]\[ \angle M = 11x = 11 \times 11 = 121^\circ \][/tex]
6. Calculate [tex]\( \angle N \)[/tex] (using [tex]\( x = 11 \)[/tex]):
[tex]\[ \angle N = 6x - 7 = 6 \times 11 - 7 = 66 - 7 = 59^\circ \][/tex]
7. Calculate the remaining angles:
[tex]\[ \angle O = \angle M = 121^\circ \quad \text{(opposite to } \angle M \text{)} \][/tex]
[tex]\[ \angle L = \angle N = 59^\circ \quad \text{(opposite to } \angle N \text{)} \][/tex]
Now that we have all the angles, we can identify which statements are true:
1. [tex]\( x = 11 \)[/tex]:
This statement is true because we solved [tex]\( x = 11 \)[/tex].
2. [tex]\( m L L = 22^\circ \)[/tex]:
This statement is false. It does not correspond to any of the angles calculated.
3. [tex]\( m_{\angle} M = 111^\circ \)[/tex]:
This statement is false. The correct measure of [tex]\( \angle M \)[/tex] is [tex]\( 121^\circ \)[/tex].
4. [tex]\( m_{\angle} N = 59^\circ \)[/tex]:
This statement is true and matches our calculation.
5. [tex]\( m_{\angle} O = 121^\circ \)[/tex]:
This statement is true because [tex]\( \angle O \)[/tex] is opposite [tex]\( \angle M \)[/tex] and equals [tex]\( 121^\circ \)[/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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