Question

name: 11. m∠abc 12. m∠abc 13. m∠def

Explanation:

1. For problem 11:

• # Explanation:
• ## Step1: Set up the equation for vertical - angles
• Since vertical angles are equal, we set \(3n - 47=n + 7\).
• Subtract \(n\) from both sides: \(3n - n-47=n - n + 7\), which simplifies to \(2n-47 = 7\).
• Add 47 to both sides: \(2n-47 + 47=7 + 47\), so \(2n=54\).
• Divide both sides by 2: \(n=\frac{54}{2}=27\).
• ## Step2: Find the measure of \(\angle ABC\)
• Substitute \(n = 27\) into the expression for \(\angle ABC\) (either \(3n - 47\) or \(n + 7\)). Using \(n + 7\), we get \(m\angle ABC=27 + 7=34^{\circ}\).
• # Answer:
• \(34^{\circ}\)

1. For problem 12:

• # Explanation:
• ## Step1: Use the property of vertical - angles
• Since vertical angles are equal, \(x = 83\). So \(m\angle ABC = 83^{\circ}\).
• # Answer:
• \(83^{\circ}\)

1. For problem 13:

• # Explanation:
• ## Step1: Set up the equation for vertical - angles
• Since vertical angles are equal, \(8x-34 = 5x + 2\).
• Subtract \(5x\) from both sides: \(8x-5x-34=5x-5x + 2\), which gives \(3x-34 = 2\).
• Add 34 to both sides: \(3x-34 + 34=2 + 34\), so \(3x=36\).
• Divide both sides by 3: \(x=\frac{36}{3}=12\).
• ## Step2: Find the measure of \(\angle DEF\)
• Substitute \(x = 12\) into the expression \(5x + 2\). Then \(m\angle DEF=5\times12 + 2=60 + 2=62^{\circ}\).
• # Answer:
• \(62^{\circ}\)

Answer:

1. For problem 11:

• # Explanation:
• ## Step1: Set up the equation for vertical - angles
• Since vertical angles are equal, we set \(3n - 47=n + 7\).
• Subtract \(n\) from both sides: \(3n - n-47=n - n + 7\), which simplifies to \(2n-47 = 7\).
• Add 47 to both sides: \(2n-47 + 47=7 + 47\), so \(2n=54\).
• Divide both sides by 2: \(n=\frac{54}{2}=27\).
• ## Step2: Find the measure of \(\angle ABC\)
• Substitute \(n = 27\) into the expression for \(\angle ABC\) (either \(3n - 47\) or \(n + 7\)). Using \(n + 7\), we get \(m\angle ABC=27 + 7=34^{\circ}\).
• # Answer:
• \(34^{\circ}\)

1. For problem 12:

• # Explanation:
• ## Step1: Use the property of vertical - angles
• Since vertical angles are equal, \(x = 83\). So \(m\angle ABC = 83^{\circ}\).
• # Answer:
• \(83^{\circ}\)

1. For problem 13:

• # Explanation:
• ## Step1: Set up the equation for vertical - angles
• Since vertical angles are equal, \(8x-34 = 5x + 2\).
• Subtract \(5x\) from both sides: \(8x-5x-34=5x-5x + 2\), which gives \(3x-34 = 2\).
• Add 34 to both sides: \(3x-34 + 34=2 + 34\), so \(3x=36\).
• Divide both sides by 3: \(x=\frac{36}{3}=12\).
• ## Step2: Find the measure of \(\angle DEF\)
• Substitute \(x = 12\) into the expression \(5x + 2\). Then \(m\angle DEF=5\times12 + 2=60 + 2=62^{\circ}\).
• # Answer:
• \(62^{\circ}\)