Geometry Fundamentals and Logic Principles

Posted on Jan 4, 2026 in Mathematics

Chapter 1: Foundations of Geometry

1.1: Points, Lines, and Planes

  • Collinear: Points on the same line.
  • Coplanar: Points that lie in the same plane.
  • Segment: A part of a line consisting of two endpoints and all points between them.
  • Endpoint: A point at the end of a segment or the starting point of a ray.
  • Ray: Part of a line that extends from one endpoint infinitely in one direction.
  • Opposite rays: Two rays that have a common endpoint and form a line.
  • Postulate: A statement that is accepted as true without proof.

1.2: Measuring and Constructing Segments

AB = |a – b| or |b – a| (So one half plus the other half equals the whole). If B is between A and C, then AB + BC = AC (Segment Addition Postulate). Midpoint: The point that bisects a segment into two congruent segments.

1.3: Measuring and Naming Angles

Angle: A figure formed by two rays, or sides, with a common endpoint called the vertex. Ways of naming: By the vertex, by a point on each ray and the vertex, or by a number. Interior of an angle: The set of all points between the sides of the angle. Exterior of an angle: The set of all points outside the angle. Angle name: You must use all three points to name the angle, and the middle point is always the vertex. Arc marks are used to show congruence between angles. If S is in the interior of ∠PQR, then m∠PQS + m∠SQR = m∠PQR (Angle Addition Postulate).

1.4: Pairs of Angles

  • Adjacent angles: Two angles in the same plane with a common vertex and a common side, but no common interior points.
  • Linear pair: A pair of adjacent angles whose non-common sides are opposite rays (they form a line).
  • Complementary: Two angles whose measures sum to 90°.
  • Supplementary: Two angles whose measures sum to 180°.

1.5: Formulas for Perimeter, Circumference, and Area

Perimeter does not get the squared sign; only area does.

  • Rectangle: P = 2l + 2w; A = lw.
  • Square: P = 4s; A = s².
  • Triangle: P = a + b + c; A = 1/2bh or bh/2.
  • Circle: The diameter passes through the center, the radius is from the center to the edge, and the circumference is the distance around the circle. r = 1/2d; d = 2r; C = πd; C = 2πr.

1.6: Coordinate Geometry

Pythagorean Theorem: a² + b² = c². Distance Formula: d = √((x&sub2; – x&sub1;)² + (y&sub2; – y&sub1;)²). Midpoint Formula: M = ((x&sub1; + x&sub2;)/2, (y&sub1; + y&sub2;)/2).

Chapter 2: Geometric Reasoning

2.1: Inductive Reasoning and Conjectures

  • Inductive reasoning: The process of reasoning that a rule or statement is true because specific cases are true.
  • Conjecture: A statement you believe to be true based on inductive reasoning.

2.2: Conditional Statements

  • Conditional: A statement that can be written in the form “If p, then q.”
  • Hypothesis: The p part of the statement following the word “if.”
  • Conclusion: The q part of the statement following the word “then.”
  • Symbol: p → q.
  • Converse: A statement formed by exchanging the hypothesis and conclusion (q → p).
  • Inverse: A statement formed by negating both the hypothesis and conclusion (~p → ~q).
  • Contrapositive: A statement formed by both exchanging and negating the hypothesis and conclusion (~q → ~p).

2.3: Deductive Reasoning

  • Deductive reasoning: The process of using logic to draw conclusions from given facts, definitions, and properties.
  • Law of Detachment: If p → q is a true statement and p is true, then q is true.
  • Law of Syllogism: If p → q and q → r are true statements, then p → r is true.

2.4: Biconditional Statements

Biconditional: p ↔ q means p → q and q → p. This means “p if and only if q” (p iff q).

2.5: Algebraic Properties of Equality (POE)

  • Addition POE: If a = b, then a + c = b + c.
  • Subtraction POE: If a = b, then a – c = b – c.
  • Multiplication POE: If a = b, then ac = bc.
  • Division POE: If a = b and c ≠ 0, then a/c = b/c.
  • Reflexive POE: a = a.
  • Symmetric POE: If a = b, then b = a.
  • Transitive POE: If a = b and b = c, then a = c.
  • Substitution POE: If a = b, then b can be substituted for a in any expression.
  • Reflexive POC (Congruence): EF ≅ EF.
  • Symmetric POC: If ∠1 ≅ ∠2, then ∠2 ≅ ∠1.
  • Transitive POC: If PQ ≅ RS and RS ≅ TU, then PQ ≅ TU.

Postulates do not require proof; theorems do.

Chapter 3: Parallel and Perpendicular Lines

3.1: Angle Relationships and Slope

Corresponding angles (1), Alternate Interior angles (2). (Draw on bottom of page). Point-slope form: y – y&sub1; = m(x – x&sub1;). Slope-intercept form: y = mx + b.

Chapter 4: Triangle Congruence

4.1: Classifying Triangles

  • Acute: A triangle with three acute angles.
  • Right: A triangle with one right angle and two acute angles.
  • Obtuse: A triangle with one obtuse angle.
  • Equiangular: A triangle where all angles are equal.

Triangle Sum Theorem: m∠A + m∠B + m∠C = 180°. Acute angles in a right triangle are complementary. The measure of each angle in an equiangular triangle is 60°.