Geometry: Undefined Terms and Subsets of a Line

• Geometry is a branch of mathematics that deals with shape, size, the relative position of figures, and the properties of space; the term originates from ancient Greek.
• A geometer is a mathematician specializing in geometry.

Undefined Terms

• Points have an exact location but no length, width, or thickness; they are dimensionless and represented by a dot and a capital letter (e.g., Pt. A or A).
• Lines have infinite length but no width or thickness, extending infinitely in two directions; they are straight and one-dimensional, and represented by a double-headed arrow, two capital letters, or a lowercase script letter.
• Planes have infinite length and width but no thickness, extending in all directions; they are flat and two-dimensional, represented by a parallelogram with a capital script letter at the corner or three consecutive vertices.

Subset of a Line

• Ray: A subset of a line with one endpoint.
• Line Segment (Segment): A subset of a line with two endpoints.
• Midpoint: A point that divides a segment into two equal parts.
• Congruent Segments: Two segments with the same measurement.
• Betweenness: A point between two other points (not necessarily in the middle).
• Opposite rays: Share a common endpoint and extend in opposite directions.
• Half-Line: All points on one side of a fixed point on a line, excluding the fixed point.

Relationships of Points, Lines, and Planes

• Collinear Points: Three or more points on the same line.
• Non-collinear Points: Three or more points not on the same line.
• Coplanar Points: Three or more points on the same plane.
• Non-Coplanar Points: Three or more points not on the same plane.
• Intersecting Lines: Two coplanar lines that intersect at a common point.
• Parallel Lines: Coplanar lines that do not intersect.
• Perpendicular Lines: Two intersecting lines that form four right angles.
• Concurrent Lines: Three or more lines that intersect at a common point.
• Skew Lines: Non-coplanar lines that do not intersect.
• Intersecting Planes: Two planes that intersect at a common line.
• Parallel Planes: Two planes that do not intersect.
• Perpendicular Planes: Two intersecting planes that form four right dihedral angles.

Conditional Statements

• Conditional Statement: A statement in "if-then" form, mathematically represented as p → q.
• Truth Value: The truth or falsity of a statement.
• Truth Table: Organizes truth values of statements.
• Theorem: A true statement demonstrated through mathematical operations.
• Postulate: A statement accepted as true without proof.
• If-then Statement: Consists of an "if" clause (p) and a "then" clause (q); "if p, then q."
• Hypothesis: The "p" statement in a conditional statement.
• Conclusion: The "q" statement in a conditional statement.
• Ways of Expressing Conditional Statements: "If p, then q," "p only if q," "q if p," "q whenever p."
• A conditional statement (p → q) is true whenever q is true, but when 𝑝 → 𝑞 is true, q is not necessarily true.

Additional Information

• Statement: An expression with a complete thought.
• To assess the truth of a conditional statement, consider the truth table: the statement is only false when the hypothesis (p) is true and the conclusion (q) is false.

Triangle

• Triangle: A polygon with three sides, formed by the union of 3 non-collinear segments intersecting at their endpoints.
  • Vertices: endpoints
  • Sides: line segments
  • Angle: union of two non-collinear segments
  • Interior: points inside the angles.
  • Exterior: points outside the triangle.
  • Included Angle: vertex is the common endpoint of the given segments
  • Included Side: endpoints are the vertices of the given angles
• Classification by angles:
  • Acute: all angles are acute.
  • Right: one right angle.
  • Obtuse: one obtuse angle.
  • Equiangular: all angles are congruent.
• Classification by congruent sides:
  • Scalene: no congruent sides.
  • Isosceles: two congruent sides.
  • Equilateral: all sides are congruent.
• Right Triangle Parts:
  • Hypotenuse: longest side, opposite the right angle.
  • Legs: sides forming the right angle.
• Isosceles Triangle Parts:
  • Legs: congruent sides.
  • Base: non-congruent side.
  • Vertex Angle: angle included by the congruent sides.
  • Base Angles: angles including the base.
• Isosceles Right Triangle: A right triangle with congruent legs.
• Pythagorean Theorem: 𝑐2 = 𝑎2 + 𝑏2 (a and b are legs, c is hypotenuse).
• Secondary Parts:
  • Altitude: segment from a vertex perpendicular to the opposite side.
  • Median: segment from a vertex to the midpoint of the opposite side.
  • Angle Bisector: segment from a vertex to a point on the opposite side.
  • Perpendicular Bisector: combination of altitude and median.

Quadrilaterals

• Quadrilaterals are polygons with four sides. Types include:
  • Parallelogram: two pairs of parallel opposite sides.
  • Trapezoid: one pair of parallel sides.
  • Trapezium/General Quadrilateral: no parallel sides.
  • Rectangle: parallelogram with four right angles.
  • Rhombus: parallelogram with four equal sides.
  • Square: parallelogram with four right angles and four equal sides.
  • Isosceles Trapezoid: trapezoid with congruent legs.