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Geometry Concepts and Relationships

Aug 23, 2024

Geometry Concepts Overview

Lines, Rays, and Segments

  • Line: Extends infinitely in both directions, represented with arrows.
    • Example: Line AB, Line BC, Line AC.
  • Ray: Has a starting point and extends infinitely in one direction, represented with a point and an arrow.
    • Example: Ray AB (starting at A).
  • Segment: Has a definite starting and ending point.
    • Example: Segment AB.

Angles

  1. Acute Angle: Measures between 0 and 90 degrees.
    • Formed by two rays (e.g., Angle ABC).
  2. Right Angle: Measures exactly 90 degrees.
  3. Obtuse Angle: Measures between 90 and 180 degrees.
  4. Straight Angle: Measures exactly 180 degrees.

Midpoints and Bisectors

  • Midpoint: The point in the middle of a segment, creating two equal segments.
    • Example: If B is the midpoint of segment AC, then segment AB ≅ segment BC.
  • Segment Bisector: A line/ray that passes through the midpoint, creating equal segments.
  • Angle Bisector: A ray that divides an angle into two equal angles.

Parallel and Perpendicular Lines

  • Parallel Lines:
    • Never intersect, have the same slope.
    • Symbol: a || b.
  • Perpendicular Lines:
    • Intersect at right angles.
    • Slopes are negative reciprocals of each other.
    • Symbol: a ⊥ b.

Angle Relationships

  • Complementary Angles: Add up to 90 degrees.
    • Example: If Angle A = 40° and Angle C = 50°, then A and C are complementary.
  • Supplementary Angles: Add up to 180 degrees.
    • Example: If Angle ABD = 110°, then Angle DBC = 70°.

Transitive Property

  • If two angles are congruent to the same angle, they are congruent to each other.
    • Example: If Angle 1 ≅ Angle 2 and Angle 3 ≅ Angle 2, then Angle 1 ≅ Angle 3.

Vertical Angles

  • Formed by two intersecting lines. Opposite angles are congruent.
    • Example: If Angle 1 = 50°, Angle 3 = 50° (vertical angles).

Medians and Altitudes

  • Median: A segment from a vertex to the midpoint of the opposite side in a triangle.
  • Altitude: A segment from a vertex perpendicular to the opposite side, forming right angles within the triangle.

Perpendicular Bisectors

  • A line that is both perpendicular and bisects a segment.
    • Example: If line L bisects segment AB at midpoint M and forms right angles.

Triangle Congruence Postulates

  1. SSS Postulate: If all three sides of one triangle are congruent to all three sides of another triangle, they are congruent.
  2. SAS Postulate: If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, they are congruent.
  3. ASA Postulate: If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, they are congruent.
  4. AAS Postulate: If two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of another triangle, they are congruent.

Additional Concepts

  • CPCTC: Corresponding Parts of Congruent Triangles are Congruent.
  • Use vertical angles and properties of altitudes when proving triangle congruence.

Practice

  • Check the description for additional practice problems and resources.
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