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1.8 – Parallel Lines and Proofs

Objectives

  • Define skew and parallel lines.
  • Use the correct notation to indicate parallel lines.
  • Use the special angle relationships created by transversals and parallel lines to determine angle measures for corresponding angles, alternate interior angles, and consecutive angles.

 

Key Terms

  • Alternate Interior Angles – two angles formed by a line (called a transversal) that intersects two parallel lines. The angles are on opposite sides of the transversal and inside the parallel lines.
  • Angle of Incidence – the angle between a ray of light meeting a surface and the line perpendicular to the surface at the point of contact.
  • Angle of Reflection – the angle between a ray of light reflecting off a surface and the line perpendicular to the surface at the point of contact.
  • Consecutive Interior Angles – two angles formed by a line (called a transversal) that intersects two parallel lines. The angles are on the same side of the transversal and are inside the parallel lines.
  • Corresponding Angles – two nonadjacent angles formed on the same side (same place) of a line (called a transversal) that intersects two parallel lines, with one angle interior and one angle exterior to the lines.
  • Intersect – to cross over one another.
  • Law of Reflection – a law stating that the angle of incidence is congruent to the angle of reflection.
  • Parallel Lines – lines lying in the same plane without intersecting. Two or more lines are parallel if they lie in the same plane and do not intersect. Notation: the symbol || means “parallel.” If parallel lines are graphed on a Cartesian coordinate system, they have the same slope.
  • Skew Lines – lines that are not in the same plane. Skew lines do not intersect, and they are not parallel.
  • Transversal – a line, ray, or segment that intersects two or more coplanar lines, rays, or segments at different points.

 

Notes

Parallel Lines
  • Two lines are parallel if they exist on the same plane and never intersect.
  • If any two corresponding angels are congruent, then the lines are parallel.
  • If any two alternate interior angles are congruent, then the lines are parallel.
  • If any two consecutive interior angles are supplementary, then the lines are parallel.

GeoA - 1.08 Parallel Lines

Skew Lines
  • One way that two lines won’t intersect is if they are not in the same plane.

GeoA - 1.08 Skew Lines

GeoA - 1.08 Parallel Skew

Transversals
  • When two lines are parallel, a transversal intersects both parallel lines, and two congruent angles are formed at the intersection.
  • Conjecture: When a transversal intersects two parallel lines, it forms two congruent angles.
  • Corresponding Angles Theorem
    • When one transversal crosses a pair of parallel lines, it creates four pairs of corresponding angles.
    • Corresponding angles are congruent.

GeoA - 1.08 Corresponding Angles

GeoA - 1.08 Corresponding2 GeoA - 1.08 Corresponding GeoA - 1.08 Corresponding

  • Alternate Interior Angles Theorem
    • They are created by a transversal crossing parallel lines.
    • They are not adjacent.
    • They are on opposite sides of the transversal.
    • They are congruent.

GeoA - 1.08 Alt Int Angles

GeoA - 1.08 Alt Int 02 GeoA - 1.08 Alt Int GeoA - 1.08 Alt Int Angles Proof

  • Consecutive Interior Angles Postulate
    • If two parallel lines are cut by a transversal, the consecutive interior angles are supplementary (add up to 180 degrees)

GeoA - 1.08 Consecutive Int Angles GeoA - 1.08 Consecutive Int

GeoA - 1.08 Consecutive Int 2

  • The law of equality: if A = B and B = C, then A = C

GeoA - 1.08 Equal Same GeoA - 1.08 Angle Pairs

Law of Reflection
  • How does light reflect in a mirror?

GeoA - 1.08 Angle of Incidence Reflection

GeoA - 1.08 Angle of Incidence Reflection Double Mirror

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