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