Parallel Lines – Angle Facts

Statement

When a transversal cuts across two parallel lines, several special angle facts arise:

• Corresponding angles are equal.
• Alternate angles are equal.
• Co-interior (allied) angles add up to \(180^{\circ}\).

These facts are fundamental in geometry proofs and angle-chasing problems.

Why it’s true

• Corresponding angles: When two parallel lines are cut by a transversal, the 'F'-shaped angle pairs are equal because the parallel lines ensure the transversal crosses at the same slope.
• Alternate angles: The 'Z'-shaped angle pairs are equal because each parallel line creates the same inclination with the transversal.
• Co-interior angles: The 'C'-shaped angle pairs add to \(180^\circ\) because they are supplementary, filling a straight line when placed together.

Recipe (how to use it)

1. Identify which type of angle pair you are working with: corresponding, alternate, or co-interior.
2. Apply the rule:
   • Corresponding → equal.
   • Alternate → equal.
   • Co-interior → sum to \(180^{\circ}\).
3. Solve for unknown angles accordingly.

Spotting it

Look for parallel line symbols (arrows on lines) and a transversal cutting across. Identify the 'F', 'Z', or 'C' shapes in the diagram — each corresponds to a specific angle rule.

Common pairings

• Triangle proofs involving parallel lines.
• Polygons with parallel sides (parallelograms, trapeziums).
• Geometry exam questions requiring angle reasons.

Mini examples

1. Given: Corresponding angle is \(65^\circ\). Find: other corresponding angle. Answer: \(65^\circ\).
2. Given: Alternate angle is \(72^\circ\). Find: other alternate angle. Answer: \(72^\circ\).
3. Given: One co-interior angle is \(110^\circ\). Find: other. Answer: \(70^\circ\).

Pitfalls

• Mixing up alternate and corresponding angles — check carefully for 'F' vs 'Z'.
• Forgetting co-interior are supplementary, not equal.
• Assuming lines are parallel without checking given information.

Exam strategy

• Always state the angle fact you are using (e.g., “alternate angles on parallel lines are equal”).
• Look for chain reasoning — often you need two or three steps using these rules.
• Remember co-interior pairs are not equal, but add to 180°.

Summary

Parallel lines and a transversal give rise to three key angle facts: corresponding angles equal, alternate angles equal, and co-interior angles supplementary. Recognising these patterns allows you to solve complex geometry problems efficiently.