Basic Concepts in Geometry

Lines, Rays, and Segments

• Line: Extends in opposite directions forever; represented with two arrows.
  • Example: Line AB, Line BC, or Line AC
• Ray: Has a starting point and extends forever in one direction; represented with an arrow.
  • Example: Ray AB (can't be BC if A is starting point)
• Segment: Has a defined start and end.
  • Example: Segment AB or Segment BA

Angles

• Acute Angle: Measure between 0 and 90 degrees
• Right Angle: Measure of 90 degrees
• Obtuse Angle: Measure greater than 90 but less than 180 degrees
• Straight Angle: Measure of 180 degrees

Midpoints and Bisectors

• Midpoint: Middle point of a segment; divides into two equal segments.
  • Example: B is midpoint of AC if AB = BC
• Segment Bisector: A line/ray that cuts a segment into two equal parts.
• Angle Bisector: A ray that divides an angle into two equal parts.

Parallel and Perpendicular Lines

• Parallel Lines: Never intersect; have the same slope.
  • Notation: A || B
• Perpendicular Lines: Intersect at right angles.
  • Notation: A ⊥ B

Complementary and Supplementary Angles

• Complementary Angles: Two angles whose measures add up to 90 degrees.
• Supplementary Angles: Two angles whose measures add up to 180 degrees.

Transitive Property

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

Vertical Angles

• Formed by two intersecting lines; opposite angles are congruent.

Medians and Altitudes in Triangles

• Median: Line segment from a vertex to the midpoint of the opposite side.
• Altitude: Perpendicular line segment from a vertex to the opposite side.

Perpendicular Bisectors

• Bisect a segment into two equal parts and form right angles with it.
• Any point on a perpendicular bisector is equidistant to the segment's endpoints.

Proving Congruence in Triangles

• SSS Postulate: Side-Side-Side
• SAS Postulate: Side-Angle-Side
• ASA Postulate: Angle-Side-Angle
• AAS Postulate: Angle-Angle-Side
• CPCTC: Corresponding Parts of Congruent Triangles are Congruent

Practice Problems and Further Learning

• For more practice, refer to recommended resources, videos, and playlists for comprehensive coverage of geometry topics and practice problems.
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