Question

5.5 angle relationships with parallel lines
decide whether the statement is true or false. if the statement is false, correct the statement.

1. ∠1 and ∠2 are vertical angles
2. ∠1 and ∠5 are corresponding angles
3. ∠2 and ∠5 are alternate exterior angles
4. ∠4 and ∠6 are alternate interior angles
5. ∠7 and ∠2 are corresponding angles
6. ∠8 and ∠2 are alternate exterior angles
7. ∠3 is congruent to ∠8
8. ∠3 and ∠7 are corresponding angles
9. ∠3 and ∠6 are interior angles on the same side of the transversal

find the value of x.

1. m∠1 = 5x - 19, m∠2 = 4x + 10
2. m∠1 = 10x - 26, m∠2 = 7x + 40
3. m∠1 = 8x - 19, m∠2 = 6x + 11
4. m∠1 = 7x - 16, m∠2 = 4x + 20
5. m∠1 = 15x + 8, m∠2 = 8x - 12
6. m∠1 = 3x + 14, m∠2 = 5x - 34

Explanation:

1. For the problem of finding the value of \(x\) in \(m\angle1 = 5x-19\) and \(m\angle2=4x + 10\):

• Explanation:
• ## Step 1: Assume \(\angle1\) and \(\angle2\) are vertical - angles (since they are opposite each other at the intersection of two lines), and vertical - angles are equal.
• Set up the equation \(5x-19=4x + 10\).
• ## Step 2: Solve the equation for \(x\).
• Subtract \(4x\) from both sides: \(5x-4x-19=4x-4x + 10\), which simplifies to \(x-19 = 10\).
• Then add 19 to both sides: \(x-19 + 19=10 + 19\), so \(x=29\).

1. For \(m\angle1 = 10x-26\) and \(m\angle2=7x + 40\):

• Explanation:
• ## Step 1: Set up the equation assuming \(\angle1\) and \(\angle2\) are vertical - angles.
• \(10x-26=7x + 40\).
• ## Step 2: Solve for \(x\).
• Subtract \(7x\) from both sides: \(10x-7x-26=7x-7x + 40\), giving \(3x-26 = 40\).
• Add 26 to both sides: \(3x-26 + 26=40 + 26\), so \(3x=66\).
• Divide both sides by 3: \(x = 22\).

1. For \(m\angle1 = 8x-19\) and \(m\angle2=6x + 11\):

• Explanation:
• ## Step 1: Set up the equation based on vertical - angle equality.
• \(8x-19=6x + 11\).
• ## Step 2: Solve for \(x\).
• Subtract \(6x\) from both sides: \(8x-6x-19=6x-6x + 11\), getting \(2x-19 = 11\).
• Add 19 to both sides: \(2x-19 + 19=11 + 19\), so \(2x=30\).
• Divide both sides by 2: \(x = 15\).

1. For \(m\angle1 = 7x-16\) and \(m\angle2=4x + 20\):

• Explanation:
• ## Step 1: Set up the equation.
• \(7x-16=4x + 20\).
• ## Step 2: Solve for \(x\).
• Subtract \(4x\) from both sides: \(7x-4x-16=4x-4x + 20\), giving \(3x-16 = 20\).
• Add 16 to both sides: \(3x-16 + 16=20 + 16\), so \(3x=36\).
• Divide both sides by 3: \(x = 12\).

1. For \(m\angle1 = 15x+8\) and \(m\angle2=6x - 12\):

• Explanation:
• ## Step 1: Set up the equation.
• \(15x+8=6x - 12\).
• ## Step 2: Solve for \(x\).
• Subtract \(6x\) from both sides: \(15x-6x+8=6x-6x - 12\), getting \(9x+8=-12\).
• Subtract 8 from both sides: \(9x+8 - 8=-12 - 8\), so \(9x=-20\).
• Divide both sides by 9: \(x=-\frac{20}{9}\).

1. For \(m\angle1 = 3x+14\) and \(m\angle2=5x - 34\):

• Explanation:
• ## Step 1: Set up the equation.
• \(3x+14=5x - 34\).
• ## Step 2: Solve for \(x\).
• Subtract \(3x\) from both sides: \(3x-3x+14=5x-3x - 34\), giving \(14 = 2x-34\).
• Add 34 to both sides: \(14 + 34=2x-34 + 34\), so \(48 = 2x\).
• Divide both sides by 2: \(x = 24\).

Answer:

1. \(x = 29\)
2. \(x = 22\)
3. \(x = 15\)
4. \(x = 12\)
5. \(x=-\frac{20}{9}\)
6. \(x = 24\)