Pairs of Lines and Angles

• Lesson 10.1 Pairs of Lines and Angles

Lines

• Parallel Lines: Coplanar lines that do not intersect are parallel lines.
• Skew Lines: Noncoplanar lines that do not intersect are skew lines.
• Parallel Planes: Coplanar planes that do not intersect are parallel planes.
• Example: In the diagram, lines m and n are parallel lines, lines m and k are skew lines, and planes T and U are parallel planes.

Symbols for Lines

• The symbol ∥ means "is parallel to."
• Example: m ∥ n (line m is parallel to line n).

Transversals

• A transversal is a line that intersects two or more coplanar lines at different points.
• Example: In the diagram, line t is a transversal that intersects lines l and m.

Angles Formed by Transversals

• Interior Angles: Angles that lie between the two lines.
• Exterior Angles: Angles that lie outside the two lines.
• Consecutive Interior Angles: Interior angles that lie on the same side of the transversal.
• Alternate Interior Angles: Interior angles that lie on opposite sides of the transversal.
• Alternate Exterior Angles: Exterior angles that lie on opposite sides of the transversal.
• Corresponding Angles: Angles that occupy the same relative position.

Angle Pair Types

• Corresponding Angles: ∠1 and ∠5, ∠2 and ∠6, ∠3 and ∠7, ∠4 and ∠8

• Alternate Interior Angles: ∠3 and ∠6, ∠4 and ∠5

• Alternate Exterior Angles: ∠1 and ∠8, ∠2 and ∠7

• Consecutive Interior Angles: ∠3 and ∠5, ∠4 and ∠6

• Lesson 10.2 Parallel Lines and Transversals

Parallel Lines and Angle Relationships

• Corresponding Angles Theorem: If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent.
• Alternate Interior Angles Theorem: If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent.
• Alternate Exterior Angles Theorem: If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent.
• Consecutive Interior Angles Theorem: If two parallel lines are cut by a transversal, then the pairs of consecutive interior angles are supplementary.

Proof Example

• Given: p ∥ q
• Prove: ∠1 ≅ ∠5
• Statements:
  • p ∥ q
  • ∠1 ≅ ∠3
  • ∠3 ≅ ∠5
  • ∠1 ≅ ∠5
• Reasons:
  • Given
  • Vertical Angles Theorem
  • Corresponding Angles Theorem
  • Transitive Property of Congruence

Using Angle Relationships

• When lines are parallel, specific angle pairs are congruent or supplementary. These relationships can be used to find unknown angle measures.

Determining Parallel Lines

• If corresponding angles are congruent, then the lines are parallel.

• If alternate interior angles are congruent, then the lines are parallel.

• If alternate exterior angles are congruent, then the lines are parallel.

• If consecutive interior angles are supplementary, then the lines are parallel.

• Lesson 10.3 Proofs with Parallel Lines

Theorems

• Corresponding Angles Converse: If two lines are cut by a transversal so the corresponding angles are congruent, then the lines are parallel.
• Alternate Interior Angles Converse: If two lines are cut by a transversal so the alternate interior angles are congruent, then the lines are parallel.
• Alternate Exterior Angles Converse: If two lines are cut by a transversal so the alternate exterior angles are congruent, then the lines are parallel.
• Consecutive Interior Angles Converse: If two lines are cut by a transversal so the consecutive interior angles are supplementary, then the lines are parallel.

Transitive Property of Parallel Lines

• If two lines are parallel to the same line, then they are parallel to each other.
• If a ∥ b and b ∥ c, then a ∥ c.

Perpendicular Transversal Theorem

• If a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other.
• If a ∥ b and t ⊥ a, then t ⊥ b.

Lines Perpendicular to a Transversal Theorem

• If two lines are perpendicular to the same line, then they are parallel.
• If a ⊥ t and b ⊥ t, then a ∥ b.

Proof Strategies

• Use given information to identify congruent or supplementary angle pairs.
• Apply the converses of the theorems to prove that two lines are parallel.
• Utilize properties of parallel lines to deduce new relationships.
• Apply the transitive property to link parallel lines through a common parallel line.

Example Proof

• Given: ∠1 ≅ ∠5
• Prove: p ∥ q
• Statements:
  • ∠1 ≅ ∠5
  • ∠3 ≅ ∠1
  • ∠3 ≅ ∠5
  • p ∥ q
• Reasons:
  • Given
  • Vertical Angles Theorem
  • Transitive Property of Congruence
  • Corresponding Angles Converse