Question 1: State the key property that defines two lines as parallel.

Answer: They are always the same distance apart and do not intersect.

Question 2: Calculate the slope of a line parallel to the line y = 3x + 2.

Answer: 3

Question 3: Construct a line parallel to y = -2x + 5 passing through the point (4, 1). Write the equation of the new line.

Answer: y = -2x + 9

Question 4: A line is equidistant from two parallel lines. What is the locus of all such points?

Answer: The line exactly halfway between the two lines.

Question 5: If two lines are parallel, and one line has equation 2x - y + 4 = 0, what is the equation of a line parallel to it passing through the point (3, 5)?

Answer: y = 2x + 1

Question 6: In a diagram, two parallel lines are cut by a transversal, creating alternate interior angles of 75°. Explain why these angles are equal and what this signifies about the lines.

Answer: Alternate interior angles are equal when lines are parallel, indicating the lines do not intersect and are always equidistant.

Question 7: Given the lines y = 2x + 1 and y = 2x - 3, find the distance between them.

Answer: 4

Question 8: A line passes through the point (2, 3) and is parallel to the line 4x - y + 6 = 0. Write its equation.

Answer: y = 4x - 5

Question 9: Describe how to construct a line that is equidistant from two parallel lines using only a compass and straightedge.

Answer: Construct the midpoint of the segment connecting the two lines at any point, then draw a line through the midpoint parallel to the lines.

Question 10: Identify and correct the mistake in the following statement: 'All parallel lines are perpendicular to the same transversal.'

Answer: Parallel lines are not necessarily perpendicular; they are simply equidistant and do not intersect.

Question 11: Challenge Question: Two lines are parallel, with equations y = 5x + 1 and y = 5x - 4. Find the coordinates of the point exactly halfway between these lines at x = 2.

Answer: (2, 6)