Question 1: In the diagram, lines l and m are parallel, and the Z-pattern is formed with angles A, B, and C. If angle A is 65°, find angles B and C.

Answer: {"B":65,"C":115}

Question 2: Construct a diagram with two parallel lines and a transversal, demonstrating the Z-pattern. Label three alternate exterior angles, with one given as 70°. Identify the other two.

Answer: {"angle1":70,"angle2":70,"angle3":110}

Question 3: A transversal cuts two parallel lines, creating a Z-pattern. If one of the alternate angles is 45°, what is the measure of the corresponding alternate exterior angle on the same side?

Answer: See full answer key

Question 4: Identify the error in the following reasoning: 'Since angles in a Z-pattern are equal, if one angle is 80°, the other in the Z is also 80°, but the exterior angle must then be 100°.'

Answer: The error is assuming the exterior angle is 100°; in reality, the exterior angle is supplementary to the interior angle and should be 100° only if the interior angle is 80°, but that depends on the specific diagram. Clarify the relationship between alternate interior and exterior angles.

Question 5: Calculate the unknown angle in the diagram: the interior angle adjacent to a 75° alternate interior angle, when lines are parallel and the Z-pattern applies.

Answer: See full answer key

Question 6: A real-world scenario: Two roads cross a railway line, forming a Z-pattern with parallel railway tracks. If the angle between the railway and one road is 60°, what are the angles where the roads intersect the railway?

Answer: {"angle1":60,"angle2":120}

Question 7: Construct a diagram showing two parallel lines and a transversal forming a Z-pattern. If one of the alternate angles is 55°, find the other two angles in the pattern.

Answer: {"angle1":55,"angle2":55,"angle3":125}

Question 8: Extend the Z-pattern concept: If the lines are not parallel, how do the corresponding angles change? Write a brief explanation.

Answer: When lines are not parallel, the Z-pattern no longer guarantees equal angles. Corresponding angles may differ, and the pattern does not hold, indicating skewed or intersecting lines instead.

Question 9: Identify the mistake: A student claims angles in a Z-pattern are always supplementary. Is this true? Explain why or why not.

Answer: This is false because in a Z-pattern, alternate interior angles are equal, not supplementary. Supplementary angles sum to 180°, but the Z-pattern angles are equal unless specified otherwise.

Question 10: Challenge: Given a diagram with parallel lines and a transversal forming a Z-pattern, prove that the alternate angles are equal using angle properties.

Answer: Since alternate interior angles are corresponding angles when lines are parallel, they are equal by the Alternate Interior Angles Theorem. The Z-pattern visually demonstrates this relationship.