📝 Geometry Notation Cheatsheet#
Master these symbols and conventions for clear, consistent geometric communication.
🔤 Basic Geometric Objects#
| Symbol | Meaning | Usage Cue |
|---|
| $A$, $B$, $C$ | Points | Capital letters for vertices |
| $\overline{AB}$ | Line segment | Distance between points A and B |
| $[AB]$ | Directed segment | Length with orientation (A to B) |
| $\overrightarrow{AB}$ | Ray | Half-line from A through B |
| $\overleftrightarrow{AB}$ | Line | Infinite line through A and B |
| $\angle ABC$ | Angle | Angle with vertex B, sides BA and BC |
| $\triangle ABC$ | Triangle | Triangle with vertices A, B, C |
| $\square ABCD$ | Quadrilateral | Quadrilateral with vertices A, B, C, D |
📐 Angle and Line Relationships#
| Symbol | Meaning | Usage Cue |
|---|
| $\parallel$ | Parallel | Lines that never meet |
| $\perp$ | Perpendicular | Lines meeting at right angles |
| $\sim$ | Similar | Same shape, different size |
| $\cong$ | Congruent | Same shape and size |
| $\equiv$ | Equivalent | Equal in measure |
| $\approx$ | Approximately equal | Close in value |
| Symbol | Meaning | Usage Cue |
|---|
| $k$ | Scale factor | Ratio of similarity or homothety |
| $r$ | Radius | Distance from center to edge |
| $R$ | Circumradius | Radius of circumscribed circle |
| $r$ | Inradius | Radius of inscribed circle |
| $s$ | Semi-perimeter | Half the perimeter: $s = \frac{a+b+c}{2}$ |
📊 Coordinate Geometry#
| Symbol | Meaning | Usage Cue |
|---|
| $(x,y)$ | Point coordinates | Horizontal, then vertical |
| $m$ | Slope | Rise over run: $m = \frac{y_2-y_1}{x_2-x_1}$ |
| $d$ | Distance | Length between two points |
| $\theta$ | Angle measure | Usually in degrees or radians |
⭕ Circle Notation#
| Symbol | Meaning | Usage Cue |
|---|
| $\odot O$ | Circle with center O | Center point in subscript |
| $\widehat{AB}$ | Arc AB | Minor arc unless specified |
| $\overline{AB}$ | Chord AB | Line segment connecting two points on circle |
| $\overrightarrow{AB}$ | Tangent at A | Line touching circle at point A |
| $P \cdot P$ | Power of point P | $PA \cdot PB$ for chords/secants |
🔢 Triangle Centers#
| Symbol | Meaning | Usage Cue |
|---|
| $G$ | Centroid | Center of mass, intersection of medians |
| $I$ | Incenter | Center of incircle, intersection of angle bisectors |
| $O$ | Circumcenter | Center of circumcircle, intersection of perpendicular bisectors |
| $H$ | Orthocenter | Intersection of altitudes |
📏 Length and Area#
| Symbol | Meaning | Usage Cue |
|---|
| $a$, $b$, $c$ | Triangle sides | Opposite angles A, B, C respectively |
| $h_a$, $h_b$, $h_c$ | Altitudes | Height to sides a, b, c |
| $m_a$, $m_b$, $m_c$ | Medians | From vertices to midpoints of opposite sides |
| $A$ | Area | Surface area of the figure |
| $P$ | Perimeter | Sum of all side lengths |
🎯 Special Notations#
| Symbol | Meaning | Usage Cue |
|---|
| $\angle ABC \equiv \angle DEF$ | Angle equality | Same measure |
| $\triangle ABC \sim \triangle DEF$ | Triangle similarity | Corresponding angles equal, sides proportional |
| $\triangle ABC \cong \triangle DEF$ | Triangle congruence | All corresponding parts equal |
| $AB \parallel CD$ | Parallel lines | Never intersect |
| $AB \perp CD$ | Perpendicular lines | Meet at right angles |
🔄 Directed Angles (AMC 12)#
| Symbol | Meaning | Usage Cue |
|---|
| $\angle ABC$ | Directed angle | Measured from BA to BC |
| $\angle ABC \equiv \angle DEF \pmod{180°}$ | Mod 180° | Angles differ by multiple of 180° |
| $\angle ABC + \angle DEF \equiv 0° \pmod{180°}$ | Supplementary | Sum to 180° |
💡 Common Patterns#
Similarity Statements#
- $\triangle ABC \sim \triangle DEF$ means $\angle A = \angle D$, $\angle B = \angle E$, $\angle C = \angle F$
- And $\frac{AB}{DE} = \frac{BC}{EF} = \frac{CA}{FD}$
Congruence Statements#
- $\triangle ABC \cong \triangle DEF$ means all corresponding parts are equal
- Order matters: $A \leftrightarrow D$, $B \leftrightarrow E$, $C \leftrightarrow F$
Angle Relationships#
- $\angle ABC = \angle CBA$ (same angle, different notation)
- $\angle ABC + \angle CBD = \angle ABD$ (angle addition)
- $\angle ABC + \angle CBA = 180°$ (linear pair)
⚠️ Common Mistakes#
- Don’t confuse $\overline{AB}$ (segment) with $AB$ (length)
- Don’t mix similarity ($\sim$) and congruence ($\cong$) symbols
- Remember that $\angle ABC$ and $\angle CBA$ are the same angle
- Be consistent with point order in similarity/congruence statements
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