Explore Westonci.ca, the leading Q&A site where experts provide accurate and helpful answers to all your questions. Our platform provides a seamless experience for finding precise answers from a network of experienced professionals. Discover in-depth answers to your questions from a wide network of professionals on our user-friendly Q&A platform.
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]
Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. Your questions are important to us at Westonci.ca. Visit again for expert answers and reliable information.