Corollaries of Parallel Lines and Transversals (Leaving Cert Mathematics): Revision Notes
📚 Revision Notes
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.
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.
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.
Worked Examples
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Example 1: Using Alternate Angles
Problem: Two parallel lines and are crossed by a transversal .
If one alternate angle measures 65°, find all other alternate angles.
Solution:
- By Corollary , alternate angles are equal.
- The alternate angle on the opposite side of the transversal also measures 65°
Answer: The alternate angles are 65°
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Example 2: Using Corresponding Angles
Problem: A transversal crosses two parallel lines, creating a corresponding angle of 70°. Find all other corresponding angles.
Solution:
- By Corollary , corresponding angles are equal.
- All corresponding angles formed by the transversal and the parallel lines measure 70°
Answer: All corresponding angles are 70°.
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.