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Geometry Brain Dump – First Semester

Formulas

  • Pythagorean Theorem: a^2 + b^2 = c^2
  • Slope-Intercept Form: y = mx + b
  • Point-Slope Form: y − y1 = m(x − x1)
  • Side-Splitter Theorem:
    • \frac{AP}{PB} = \frac{AQ}{QC}
    • \frac{PB}{AB} = \frac{QC}{AC}
  • Geometric Mean Altitude Theorem: \frac{HC}{HA} = \frac{HA}{HB} \implies HC \cdot HB = (HA)^2
  • Geometric Mean Leg Theorem: \frac{HB}{BA} = \frac{BA}{BC} \implies HB \cdot BC = (BA)^2

Angle Relationships

  • Linear Pair: Sum of 180^\circ, forms straight line
  • Supplementary: Sum of 180^\circ
  • Complementary: Sum of 90^\circ
  • Acute: Less than 90^\circ
  • Right: 90^\circ
  • Obtuse: In between 90^\circ and 180^\circ
  • Straight: 180^\circ
  • Vertical: Congruent, equal measures
  • Alternate Interior: Congruent, equal measures
  • Alternate Exterior: Congruent, equal measures
  • Corresponding: Congruent, equal measures
  • Same Side (Consecutive) Interior: Sum of 180^\circ
  • Same Side (Consecutive) Exterior: Sum of 180^\circ

Rotation Rules

  • 90^\circ Clockwise or 270^\circ Counterclockwise: (x, y) \rightarrow (y, -x)
  • 90^\circ Counterclockwise or 270^\circ Clockwise: (x, y) \rightarrow (-y, x)
  • 180^\circ: (x, y) \rightarrow (-x, -y)

Reflection Rules

  • Over x-axis: (x, y) \rightarrow (x, -y)
  • Over y-axis: (x, y) \rightarrow (-x, y)
  • Over y = x: (x, y) \rightarrow (y, x)
  • Over y = -x: (x, y) \rightarrow (-y, -x)

Dilation Rule

  • Dilation with respect to the origin and scale factor of k: (x, y) \rightarrow (kx, ky)

Triangle Types

  • Right Triangle: One right angle
  • Acute Triangle: Three acute angles
  • Obtuse Triangle: One obtuse angle
  • Equiangular Triangle: All 60^\circ angles
  • Isosceles Triangle: Two congruent sides/angles
  • Scalene Triangle: No congruent sides/angles
  • Equilateral Triangle: Three congruent sides

Triangle Congruence

  • Side-Side-Side (SSS)
  • Side-Angle-Side (SAS)
  • Angle-Side-Angle (ASA)
  • Angle-Angle-Side (AAS)
  • Hypotenuse-Leg (HL)

Triangle Similarity

  • Angle-Angle (AA~)
  • Side-Side-Side (SSS~)
  • Side-Angle-Side (SAS~)

Transformations

  • Rigid Transformations (Preserves Distance): Translations, Reflections, Rotations
  • Non-Rigid Transformation (Does Not Preserve Distance if scale factor is not 1): Dilations

Conditional Statements (Given p \rightarrow q)

  • Converse: q \rightarrow p
  • Inverse: \sim p \rightarrow \sim q
  • Contrapositive: \sim q \rightarrow \sim p
  • Biconditional: p \leftrightarrow q