Question

1. converse:

(4x - 13)°
(2x + 37)°

1. converse:

(24x + 9)°
(7x - 15)°

1. converse:

(10x + 17)°
87°
(5x)°

1. converse:

(16x)°
48°
(28x)°

Explanation:

1. Question 10:

• Explanation:
• ## Step 1: Set up the equation based on corresponding - angles postulate
• If two lines \(l\) and \(m\) are parallel, then the corresponding angles are equal. So, we set \(4x - 13=2x + 37\).
• Subtract \(2x\) from both sides: \(4x-2x - 13=2x-2x + 37\), which simplifies to \(2x-13 = 37\).
• Add 13 to both sides: \(2x-13 + 13=37 + 13\), giving \(2x=50\).
• Divide both sides by 2: \(x = 25\).
• The converse is: If corresponding angles are equal, then the two lines are parallel.
• Answer: \(x = 25\), Converse: If corresponding angles are equal, then the two lines are parallel.

1. Question 11:

• Explanation:
• ## Step 1: Set up the equation based on alternate - interior angles postulate
• If two lines \(l\) and \(m\) are parallel, then the alternate - interior angles are equal. So, \(24x + 9=7x-15\).
• Subtract \(7x\) from both sides: \(24x-7x + 9=7x-7x-15\), which gives \(17x + 9=-15\).
• Subtract 9 from both sides: \(17x+9 - 9=-15 - 9\), resulting in \(17x=-24\).
• Divide both sides by 17: \(x=-\frac{24}{17}\).
• The converse is: If alternate - interior angles are equal, then the two lines are parallel.
• Answer: \(x =-\frac{24}{17}\), Converse: If alternate - interior angles are equal, then the two lines are parallel.

1. Question 12:

• Explanation:
• ## Step 1: Use the property of vertical angles and linear - pair or angle - sum property
• First, note that the angle vertical to \(87^{\circ}\) is also \(87^{\circ}\).
• If the lines \(l\) and \(m\) are parallel, then \(5x+(10x + 17)=87\) (using the property of corresponding angles or a related angle - relationship).
• Combine like terms: \(15x+17 = 87\).
• Subtract 17 from both sides: \(15x+17 - 17=87 - 17\), getting \(15x = 70\).
• Divide both sides by 15: \(x=\frac{70}{15}=\frac{14}{3}\).
• The converse is: If corresponding angles (or related angle - relationships based on parallel lines) are equal, then the two lines are parallel.
• Answer: \(x=\frac{14}{3}\), Converse: If corresponding angles (or related angle - relationships based on parallel lines) are equal, then the two lines are parallel.

1. Question 13:

• Explanation:
• ## Step 1: Use the angle - sum property and parallel - line angle relationships
• The sum of the angles in the triangle formed by the transversal and the two lines \(l\) and \(m\) and the fact that if the lines are parallel, certain angle relationships hold.
• First, the non - labeled angle in the triangle with angles \(16x\) and \(48^{\circ}\) and the angle adjacent to \(28x\) (on the same side of the transversal) are related.
• If the lines are parallel, we know that \(16x+48^{\circ}\) and \(28x\) are supplementary (same - side interior angles). So, \(16x + 48+28x=180\).
• Combine like terms: \(44x+48 = 180\).
• Subtract 48 from both sides: \(44x+48 - 48=180 - 48\), giving \(44x = 132\).
• Divide both sides by 44: \(x = 3\).
• The converse is: If same - side interior angles are supplementary, then the two lines are parallel.
• Answer: \(x = 3\), Converse: If same - side interior angles are supplementary, then the two lines are parallel.

Answer:

1. Question 10:

• Explanation:
• ## Step 1: Set up the equation based on corresponding - angles postulate
• If two lines \(l\) and \(m\) are parallel, then the corresponding angles are equal. So, we set \(4x - 13=2x + 37\).
• Subtract \(2x\) from both sides: \(4x-2x - 13=2x-2x + 37\), which simplifies to \(2x-13 = 37\).
• Add 13 to both sides: \(2x-13 + 13=37 + 13\), giving \(2x=50\).
• Divide both sides by 2: \(x = 25\).
• The converse is: If corresponding angles are equal, then the two lines are parallel.
• Answer: \(x = 25\), Converse: If corresponding angles are equal, then the two lines are parallel.

1. Question 11:

• Explanation:
• ## Step 1: Set up the equation based on alternate - interior angles postulate
• If two lines \(l\) and \(m\) are parallel, then the alternate - interior angles are equal. So, \(24x + 9=7x-15\).
• Subtract \(7x\) from both sides: \(24x-7x + 9=7x-7x-15\), which gives \(17x + 9=-15\).
• Subtract 9 from both sides: \(17x+9 - 9=-15 - 9\), resulting in \(17x=-24\).
• Divide both sides by 17: \(x=-\frac{24}{17}\).
• The converse is: If alternate - interior angles are equal, then the two lines are parallel.
• Answer: \(x =-\frac{24}{17}\), Converse: If alternate - interior angles are equal, then the two lines are parallel.

1. Question 12:

• Explanation:
• ## Step 1: Use the property of vertical angles and linear - pair or angle - sum property
• First, note that the angle vertical to \(87^{\circ}\) is also \(87^{\circ}\).
• If the lines \(l\) and \(m\) are parallel, then \(5x+(10x + 17)=87\) (using the property of corresponding angles or a related angle - relationship).
• Combine like terms: \(15x+17 = 87\).
• Subtract 17 from both sides: \(15x+17 - 17=87 - 17\), getting \(15x = 70\).
• Divide both sides by 15: \(x=\frac{70}{15}=\frac{14}{3}\).
• The converse is: If corresponding angles (or related angle - relationships based on parallel lines) are equal, then the two lines are parallel.
• Answer: \(x=\frac{14}{3}\), Converse: If corresponding angles (or related angle - relationships based on parallel lines) are equal, then the two lines are parallel.

1. Question 13:

• Explanation:
• ## Step 1: Use the angle - sum property and parallel - line angle relationships
• The sum of the angles in the triangle formed by the transversal and the two lines \(l\) and \(m\) and the fact that if the lines are parallel, certain angle relationships hold.
• First, the non - labeled angle in the triangle with angles \(16x\) and \(48^{\circ}\) and the angle adjacent to \(28x\) (on the same side of the transversal) are related.
• If the lines are parallel, we know that \(16x+48^{\circ}\) and \(28x\) are supplementary (same - side interior angles). So, \(16x + 48+28x=180\).
• Combine like terms: \(44x+48 = 180\).
• Subtract 48 from both sides: \(44x+48 - 48=180 - 48\), giving \(44x = 132\).
• Divide both sides by 44: \(x = 3\).
• The converse is: If same - side interior angles are supplementary, then the two lines are parallel.
• Answer: \(x = 3\), Converse: If same - side interior angles are supplementary, then the two lines are parallel.