Essential Geometry Concepts: Formulas, Proofs, and Theorems

Posted on Oct 10, 2025 in Mathematics

Fundamentals of Geometry

Undefined Terms and Basic Figures

  • Point: Exact location in space with no size, written as a capital letter (e.g., A).
  • Line: Straight path extending infinitely in both directions, no thickness, named with two points ($\overleftrightarrow{AB}$) or a lowercase script letter.
  • Plane: Flat surface extending infinitely in all directions, named with a script capital letter or three non-collinear points.
  • Collinear Points: Points that lie on the same line.
  • Coplanar Points: Points that lie on the same plane.
  • Segment: Part of a line with two endpoints, written $\overline{AB}$.
  • Ray: Part of a line starting at one endpoint and extending forever, written $\overrightarrow{AB}$ with an arrow.
  • Opposite Rays: Two rays sharing an endpoint that form a straight line.

Connections to Geometric Reasoning

Points, lines, and planes are the building blocks of all geometric reasoning and appear in triangle congruence proofs, circle chords, and coordinate constructions.

Angle Relationships

  • Angles: Formed by two rays (sides) meeting at a vertex.
  • Types:
    • Acute ($0^{\circ} < x < 90^{\circ}$)
    • Right ($x = 90^{\circ}$)
    • Obtuse ($90^{\circ} < x < 180^{\circ}$)
    • Straight ($x = 180^{\circ}$)
  • Adjacent Angles: Share a side and a vertex.
  • Vertical Angles: Opposite angles formed by intersecting lines; they are always congruent.
  • Linear Pair: Two adjacent angles whose non-common sides form a straight line; they are supplementary.
  • Complementary: Angles adding to $90^{\circ}$.
  • Supplementary: Angles adding to $180^{\circ}$.

Angle Connections and Shortcuts

Angle types are used in proofs, parallel line problems, and later in trigonometry for special triangles.

  • Shortcut (Vertical Angles): Look for X-shaped diagrams.
  • Shortcut (Linear Pair): Look for two angles forming a straight line.
  • Shortcut (Complementary): Check for right angle splits.

Logic, Postulates, and Proofs

Foundations of Proof

  • Postulates: Accepted facts without proof.
    • Through any two points is exactly one line.
    • Through any three non-collinear points is exactly one plane.
    • A line contains infinitely many points.
    • Two planes intersect in a line.
    • Two lines intersect in a point.
  • Theorems: Proven facts.
    • Vertical Angles Theorem: Vertical angles are congruent.
    • Right Angle Congruence Theorem: All right angles are congruent.

Segment and Angle Addition

  • Segment Addition Postulate: If B is between A and C, then $AB + BC = AC$.
  • Angle Addition Postulate: If D is in $\angle ABC$, then $m\angle ABD + m\angle DBC = m\angle ABC$.
  • Length on a Number Line: Subtract coordinates and take the absolute value: $|x_2 – x_1|$.
  • Midpoint Formula (1D): $(x_1 + x_2)/2$.

Connections and Test Traps

Segment addition is used in proving congruence and finding perimeters; midpoints are critical for bisectors, medians in triangles, and centroids.

  • Test Trap: Never assume diagrams are to scale unless stated.
  • Tip: Congruence marks (tick marks and arcs) mean equal segments or angles. Always verify relationships before using them in proofs.

Logical Statements and Reasoning

  • Conjecture: Educated guess based on observed patterns.
  • Counterexample: One example that proves a conjecture false.
  • Conditional Statement: “If p, then q” ($p \rightarrow q$).
  • Converse: “If q, then p”.
  • Inverse: “If not p, then not q”.
  • Contrapositive: “If not q, then not p”. (Logically equivalent to the conditional).
  • Biconditional: “p if and only if q” ($p \leftrightarrow q$), true only if both conditional and converse are true.
  • Deductive Reasoning: Uses facts, postulates, and definitions to make logical conclusions (backbone of proofs).
  • Inductive Reasoning: Identifies patterns and makes generalizations (used to form conjectures).

Types of Geometry and Proofs

  • Euclidean Geometry: Flat-plane geometry where parallel lines never meet and triangle angle sums equal $180^{\circ}$.
  • Spherical Geometry: Curved-surface geometry where parallel lines do not exist and triangle angle sums are greater than $180^{\circ}$.
  • Proof Types: Two-column (statements and reasons), paragraph, flowchart, and coordinate proofs.

Coordinate Geometry Essentials

Formulas for Measurement

  • Distance Formula: $d = \sqrt{((x_2 – x_1)^2 + (y_2 – y_1)^2)}$
    • Key Uses: Prove segments congruent, check if a point lies on a circle, calculate perimeters.
    • Shortcut: Subtraction order doesn’t matter because of squaring; always square root last.
  • Midpoint Formula (2D): $M = (\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2})$
    • Key Uses: Prove bisectors, locate triangle medians, find centroids.
    • Shortcut: Average x’s and y’s separately.
  • Slope Formula: $m = \frac{y_2 – y_1}{x_2 – x_1}$ (Measures steepness)
    • Parallel Lines: Same slope ($m_1 = m_2$).
    • Perpendicular Lines: Slopes are negative reciprocals ($m_1 \cdot m_2 = -1$).
    • Shortcut: Vertical lines have undefined slope; horizontal lines have slope 0.
  • Partition Formula (Ratio m:n): $(\frac{n(x_1) + m(x_2)}{m + n}, \frac{n(y_1) + m(y_2)}{m + n})$
    • Key Uses: Divide segments proportionally, locate centroids using 2:1 ratio.

Line Equations and Strategy

  • Slope-Intercept Form: $y = mx + b$ (Uses slope $m$ and y-intercept $b$).
  • Point-Slope Form: $y – y_1 = m(x – x_1)$ (Used to write line equations quickly).
  • Strategy: Use distance to show congruent sides, slope to prove parallel or perpendicular sides, and midpoint to prove bisectors.

Coordinate Proof Test Traps

  • Do not assume diagrams are drawn to scale.
  • Always calculate slope and distance directly.
  • For partition problems, double-check if the ratio is from A to B or B to A.
  • Perpendicular slopes require flipping and negating.
  • If asked for the midpoint of a horizontal segment, average x’s only; for a vertical segment, average y’s only.

Geometric Transformations

Rigid Transformations (Preserve Congruence)

Rigid transformations preserve size, shape, and distance.

  • Translation (Slide): Shifts a figure without turning or flipping it.
    • Rule: $(x, y) \rightarrow (x + a, y + b)$.
  • Reflection (Flip): Flips a figure over a line of reflection; reverses orientation.
    • 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 $y = -x$: $(x, y) \rightarrow (-y, -x)$
  • Rotation (Turn): Turns a figure around a fixed point (usually the origin).
    • $90^{\circ}$ CCW: $(x, y) \rightarrow (-y, x)$
    • $180^{\circ}$: $(x, y) \rightarrow (-x, -y)$
    • $270^{\circ}$ CCW: $(x, y) \rightarrow (y, -x)$

Non-Rigid Transformations (Similarity)

  • Dilation (Resize): Enlarges or reduces a figure relative to a center point; changes size but preserves shape.
    • Rule: $(x, y) \rightarrow (kx, ky)$, where $k$ is the scale factor.
    • $k > 1$ enlarges; $0 < k < 1$ shrinks.

Symmetry and Composition

  • Reflectional Symmetry: Figure maps onto itself after a reflection.
  • Rotational Symmetry: Figure maps onto itself after a rotation of less than $360^{\circ}$.
  • Order of Rotational Symmetry: Number of times a figure maps onto itself during a $360^{\circ}$ turn.

Transformation Test Traps

  • Check whether transformations preserve size (rigid) or only shape (non-rigid).
  • Composition of transformations is not always commutative (order matters).
  • For multiple transformations, apply them one step at a time and write intermediate coordinates.

Constructions and Triangle Properties

Compass and Straightedge Techniques

Constructions are precise drawings using only a compass and straightedge (no measurements).

  • Congruent Segments: Set compass width to the given segment and transfer.
  • Congruent Angles: Use arcs to measure the angle opening and replicate it.
  • Perpendicular Bisector: Draw arcs from both endpoints above and below the segment; connect intersections.
  • Angle Bisector: Draw an arc from the vertex; from the arc intersections, draw two arcs inside the angle; connect the vertex to the final intersection.
  • Parallel Line through a Point: Draw a transversal and copy alternate interior angles.

Triangle Theorems and Congruence

  • Triangle Inequality Theorem: The sum of any two sides of a triangle must be greater than the third side.
  • Triangle Congruence Theorems:
    • SSS (Side-Side-Side)
    • SAS (Side-Angle-Side, angle must be included)
    • ASA (Angle-Side-Angle, side must be included)
    • AAS (Angle-Angle-Side)
    • HL (Hypotenuse-Leg, for right triangles only)
  • CPCTC: Corresponding Parts of Congruent Triangles are Congruent.
  • Midsegment Theorem: Segment connecting midpoints of two sides is parallel to the third side and half its length.
  • Isosceles Triangle Theorem: Base angles of an isosceles triangle are congruent (and converse).
  • Triangle Sum Theorem: Interior angles of any triangle add to $180^{\circ}$.
  • Exterior Angle Theorem: An exterior angle equals the sum of its two remote interior angles.
  • Perpendicular Bisector Theorem: Any point on a perpendicular bisector is equidistant from the segment’s endpoints.

Similarity, Right Triangles, and Trigonometry

Similarity Criteria and Proportionality

  • Similarity Definition: Two figures are similar if corresponding angles are congruent and corresponding sides are proportional.
  • Similarity Theorems:
    • AA Similarity: Two pairs of congruent angles prove triangles are similar.
    • SSS Similarity: Corresponding side lengths are proportional.
    • SAS Similarity: One pair of congruent angles with proportional sides around it proves similarity.
  • Triangle Proportionality Theorem: If a line parallel to one side of a triangle intersects the other two sides, it divides them proportionally (and converse).
  • Angle Bisector Theorem: An angle bisector divides the opposite side into two segments proportional to the adjacent sides.

Right Triangle Relationships

  • Pythagorean Theorem: $a^2 + b^2 = c^2$, where $a$ and $b$ are legs and $c$ is the hypotenuse (and converse).
  • Special Right Triangles:
    • $45^{\circ}-45^{\circ}-90^{\circ}$: Legs are equal; hypotenuse = $\text{leg} \cdot \sqrt{2}$.
    • $30^{\circ}-60^{\circ}-90^{\circ}$: Short leg, long leg = $\text{short leg} \cdot \sqrt{3}$, hypotenuse = $2 \cdot \text{short leg}$.
  • Geometric Mean Relationships: In a right triangle with an altitude to the hypotenuse:
    • Altitude$^2$ = product of hypotenuse segments.
    • Each leg is the geometric mean of the hypotenuse and its adjacent segment.

Trigonometric Ratios

  • SOHCAHTOA:
    • Sine ($\sin \theta$) = Opposite / Hypotenuse
    • Cosine ($\cos \theta$) = Adjacent / Hypotenuse
    • Tangent ($\tan \theta$) = Opposite / Adjacent
  • Inverse Trig: Used to find angles ($\sin^{-1}$, $\cos^{-1}$, $\tan^{-1}$).

Area, Volume, and 3D Geometry

Cross-Sections and Solids of Revolution

  • Cross-Sections: Slicing a 3D figure with a plane to reveal a 2D shape.
    • Prism (Parallel to base $\rightarrow$ base shape; Perpendicular $\rightarrow$ rectangle).
    • Cylinder (Parallel to base $\rightarrow$ circle; Perpendicular $\rightarrow$ rectangle).
    • Cone (Parallel to base $\rightarrow$ smaller circle; Perpendicular $\rightarrow$ triangle).
    • Sphere (Any slice $\rightarrow$ circle).
  • Solids of Revolution: Rotating a 2D shape around an axis forms a 3D solid (e.g., rectangle $\rightarrow$ cylinder).

Dimensional Changes (Scaling)

If a linear dimension is scaled by factor $k$:

  • Perimeter: Scales by $k$ ($P_{new} = k \cdot P_{original}$).
  • Area: Scales by $k^2$ ($A_{new} = k^2 \cdot A_{original}$).
  • Volume: Scales by $k^3$ ($V_{new} = k^3 \cdot V_{original}$).

Calculating Area and Volume

  • Area of Regular Polygons: $A = (1/2) \cdot \text{Perimeter} \cdot \text{Apothem}$. (Units are square units, e.g., $\text{cm}^2$).
  • Composite Area: Break the figure into basic shapes and add/subtract areas.

Surface Area Formulas

  • Prism SA: $2B + Ph$ ($B$ = base area, $P$ = base perimeter, $h$ = height).
  • Cylinder SA: $2\pi r^2 + 2\pi rh$.
  • Pyramid SA: $B + (1/2)Pl$ ($l$ = slant height).
  • Cone SA: $\pi r^2 + \pi rl$.
  • Sphere SA: $4\pi r^2$.

Volume Formulas

(Units are cubic units, e.g., $\text{cm}^3$).

  • Prism V: $Bh$.
  • Cylinder V: $\pi r^2 h$.
  • Pyramid V: $(1/3)Bh$.
  • Cone V: $(1/3)\pi r^2 h$.
  • Sphere V: $(4/3)\pi r^3$.

Circles and Probability

Circle Terminology and Angles

  • Center: Fixed point. Radius ($r$): Distance from center to edge. Diameter: $2r$.
  • Chord: Segment with endpoints on the circle.
  • Secant: Line cutting the circle at two points.
  • Tangent: Line touching the circle at exactly one point; perpendicular to the radius at the point of contact.
  • Central Angle: Vertex at the center; measure equals the intercepted arc measure.
  • Inscribed Angle: Vertex on the circle; measure = $1/2$ the intercepted arc measure.
  • Arc Length: $L = (\theta/360) \cdot 2\pi r$ ($\theta$ in degrees).
  • Sector Area: $A = (\theta/360) \cdot \pi r^2$ ($\theta$ in degrees).

Circle Theorems and Equations

  • Power of a Point (Chords): If two chords intersect, $AE \cdot EB = CE \cdot ED$.
  • Tangent-Secant Theorem: $\text{Tangent}^2 = \text{External Part} \cdot \text{Total Secant}$ ($T^2 = E \cdot (E + I)$).
  • Equation of a Circle (Standard Form): $(x – h)^2 + (y – k)^2 = r^2$, where center = $(h, k)$ and radius = $r$.

Probability Fundamentals

  • Probability: $P = \text{favorable outcomes} / \text{total outcomes}$.
  • Complement Rule: $P(\text{not } A) = 1 – P(A)$.
  • Geometric Probability: $P = \text{area of favorable region} / \text{area of total region}$.
  • Independent Events: $P(A \text{ and } B) = P(A) \cdot P(B)$.
  • Conditional Probability: $P(A|B) = P(A \text{ and } B) / P(B)$.

Counting Principles

  • Permutations: Order matters. $P(n, r) = n! / (n – r)!$.
  • Combinations: Order does not matter. $C(n, r) = n! / (r!(n – r)!)$.