Basic Concepts in Geometry

Lines

• Definition: A line extends in opposite directions forever and is represented with two arrows.
• Naming: With points on the line, it can be named in various ways (e.g., Line AB, Line BC).

Rays

• Definition: A ray has a starting point and extends forever in one direction.
• Naming: The first point must be the starting point of the ray (e.g., Ray AB, not Ray BC).

Segments

• Definition: A segment has a beginning and an end.
• Naming: Can be represented as Segment AB, without arrows.

Angles

• Acute Angle: Measures between 0 and 90 degrees.
• Right Angle: Measures exactly 90 degrees.
• Obtuse Angle: Measures greater than 90 but less than 180 degrees.
• Straight Angle: Measures exactly 180 degrees.

Midpoint

• Definition: The midpoint is the point in the middle of a segment, creating two equal parts.
• Example: If B is the midpoint of Segment AC, then Segment AB is congruent to Segment BC.

Segment Bisector

• Definition: A ray or line that passes through the midpoint of a segment, dividing it into two equal parts.

Angle Bisector

• Definition: A ray that divides an angle into two equal parts.

Parallel Lines

• Properties: Never intersect; have the same slope.
• Notation: Line A is parallel to Line B (symbol: ||).

Perpendicular Lines

• Properties: Intersect at a right angle.
• Finding Slopes: Slopes of perpendicular lines are negative reciprocals of each other.

Complementary Angles

• Definition: Two angles that add up to 90 degrees.

Supplementary Angles

• Definition: Two angles that add up to 180 degrees.

Transitive Property

• Definition: If two angles are congruent to the same angle, they are congruent to each other.

Vertical Angles

• Definition: Opposite angles formed by intersecting lines are congruent.

Medians

• Definition: A line segment from a vertex of a triangle to the midpoint of the opposite side.

Altitudes

• Definition: A line segment from a vertex of a triangle, perpendicular to the opposite side.

Perpendicular Bisectors

• Definition: A line that is perpendicular to a segment and bisects it into two congruent parts.
• Properties: Any point on the perpendicular bisector is equidistant from the endpoints of the segment.

Congruent Triangles

• Postulates:
  • SSS (Side-Side-Side): If all three sides of two triangles are congruent, the triangles are congruent.
  • SAS (Side-Angle-Side): If two sides and the included angle of two triangles are congruent, the triangles are congruent.
  • ASA (Angle-Side-Angle): If two angles and the included side of two triangles are congruent, the triangles are congruent.
  • AAS (Angle-Angle-Side): If two angles and a non-included side of two triangles are congruent, the triangles are congruent.
  • CPCTC: Corresponding Parts of Congruent Triangles are Congruent.

Proofs & Examples

• Vertical Angles Example: If given angles are congruent, use CPCTC to prove parts of triangles are congruent.
• Using Altitudes in Proofs: Altitudes form right angles and can help prove triangle congruence using ASA or AAS postulates.

---

• Note: Additional practice problems and resources are available through the given links for further study.