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1.7 – Intersecting Lines and Proofs

Objectives

  • Define vertical angles and linear pairs.
  • Identify vertical angles and their angle measures, given a diagram and related angle measures.
  • Prove that vertical angles are congruent.
  • Define and identify perpendicular rays, lines, and segments.
  • Define perpendicular bisector.

 

Key Terms

  • Distance – the shortest and most direct path between a point and a line.
  • Perpendicular Bisector – a line, ray, or line segment that bisects a line segment at a right angle.
  • Perpendicular Lines – lines that meet to form a right angle. The symbol ⊥ means “perpendicular to.” If perpendicular lines are graphed on a Cartesian coordinate system, their slopes are negative reciprocals.
  • Vertical Angles – a pair of opposite angles formed by intersecting lines. Vertical angles have equal measures.

Notes

Review
  • Linear pair: Two adjacent (side by side) angles that form a line (add up to 180°).
  • Supplementary angles: Two angles whose measures add up to 180°.
    • They do not have to be adjacent.
  • Complementary angles: Two angles whose measures add up to 90°.
    • They do not have to be adjacent.
  • A theorem is a statement that has already been proven, so it is always true.
  • Congruent angles have equal measures
    • Equal measures does not mean equal angles.
      • Some congruent angles are equal, but ONLY IF they are the SAME exact angle.
      • Most congruent angles are NOT equal because they are two separate angles.
Linear Pairs
  • When two lines intersect, they create four linear pairs.

Vertical Angles
  • When two lines intersect, they form two pairs of vertical, congruent angles.
  • We say vertical angles are congruent because they have the same measure.
  • We do NOT say they are equal, because they are NOT the same angle.
  • Vertical angles MUST be congruent.
  • Vertical angles MUST have the same vertex.

GeoA 01.07 Vertical Angles


  • Example 1

  • Example 2

GeoA 01.07 Calculate Vertical Angles


  • Example 3

GeoA 01.07 Calculate Vertical Angles2


  • Vertical Angles Theorem

GeoA 01.07 Vertical Angle Theorem


  • Vertical Angles Proof
    • The following are REASONS (right-hand column) for proving that vertical angles are congruent.
  1. Given
  2. Definition of linear pair
  3. Subtraction
  4. Definition of linear pair
  5. Substitution
  6. Simplify
  7. Definition of congruent angles
  • Watch the video to better understand how vertical angles are proven to be congruent.
    • Notice how the simplified reasons (above) match the detailed reasons in the proof (below).

Perpendicular Lines
  • Lines, segments, or rays that cross at 90° angles
  • Perpendicular lines have a special notation. It looks like an upside-down “T.”
  • Vertical angles are congruent (each 90°) and supplementary (the pair adds up to 180°).
  • If two lines intersect to form four right angles, then the lines are perpendicular.

GeoA 01.07 Perpendicular Notation


  • A perpendicular bisector
    • A line, ray, or segment
      • The bisector is like a knife.  It cuts a segment in half.
    • Intersects a segment at its midpoint.
    • Forms right angles.
    • Divides a segment in half
      • Remember, you cannot divide a line or ray in half because they are infinite.  We can’t measure the middle of infinity.
      • Given any line \overleftrightarrow{AB} and any point C, there is exactly one line that is both perpendicular to \overleftrightarrow{AB} and passes through C.
      • Bisected segments are measurable, therefore equal to one another in length (below: \overline{AC}=\overline{CB}).

GeoA 01.07 Perpendicular Bisectors

 

  • The midpoint of a segment divides a segment in half — into two congruent segments (above, point C is the midpoint).
    • So, the perpendicular bisector divides a segment into two congruent (equal measure) segments.

 

  • Perpendicular Path Rule: For any line \overleftrightarrow{AB} and point P not on \overleftrightarrow{AB} the shortest path between P and \overleftrightarrow{AB} is a segment perpendicular to \overleftrightarrow{AB} through P.
    • In other words, the shortest path between a point and a line is a path that is perpendicular to the line.

  • Example

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