Corollaries of Parallel Lines and Transversals

Overview

When two parallel lines are crossed by a transversal, specific angle relationships are created. These relationships are fundamental in geometry, and they lead to several important corollaries regarding alternate and corresponding angles.

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Corollary 3: Alternate Angles

• Statement: Alternate angles are equal when two lines are parallel and a transversal crosses them.
• Why It Works:
  • Alternate angles lie on opposite sides of the transversal but within the bounds of the parallel lines.
  • The parallel lines ensure consistent spacing and alignment, making the alternate angles congruent.

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Corollary 4: Corresponding Angles

• Statement: Corresponding angles are equal when two lines are parallel and a transversal crosses them.
• Why It Works:
  • Corresponding angles lie on the same side of the transversal, with one angle above and one angle below the parallel lines.
  • The parallelism of the lines ensures equal spacing, leading to equal corresponding angles.

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Worked Examples

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Summary

• Corollary 3: Alternate angles are equal when two lines are parallel and a transversal crosses them.
• Corollary 4: Corresponding angles are equal when two lines are parallel and a transversal crosses them.
• These corollaries are key to solving problems involving parallel lines and transversals in geometry.