Geometry – Complete Reference Guide

Foundations

Basic Elements

Point: Exact location (0 dim).

Line: Straight path, extends infinite (1 dim).

Plane: Flat surface, extends infinite (2 dim).

Segment: Part of line with 2 endpoints.

Ray: 1 endpoint, extends infinite 1 way.

Formulas (Coordinate)

Distance: $d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$

Midpoint: $M = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$

Angles & Lines

Angle Relationships

Complementary: Sum to $90^\circ$.

Supplementary: Sum to $180^\circ$.

Vertical Angles: Opposite angles are equal.

Linear Pair: Adj angles on line sum to $180^\circ$.

Parallel Lines & Transversals

If lines are parallel:

Alt Interior Angles are Equal

Alt Exterior Angles are Equal

Corresponding Angles are Equal

Consec Interior sum to $180^\circ$

Triangles

Properties

Angle Sum: $180^\circ$

Exterior Angle: Equals sum of remote interiors.

Triangle Inequality: $a+b > c$ (Sum of any 2 > 3rd).

Congruence

CPCTC: Corr Parts of Congruent Triangles are Congruent.

Similarity

Same shape, different size.

Angles are equal, sides are proportional.

Tests: AA~, SAS~, SSS~

Segments

Median: Vertex to midpoint.

Altitude: Vertex $\perp$ to opposite side.

Midsegment: Connects midpoints. Parallel to base, 1/2 size.

Right Triangles

Pythagorean Theorem

$a^2 + b^2 = c^2$ (c is hypotenuse)

Common Triples: (3,4,5), (5,12,13), (8,15,17)

Acute if $c^2 < a^2+b^2$

Obtuse if $c^2 > a^2+b^2$

Special Right Triangles

Quadrilaterals

Sum of Interior Angles = $360^\circ$

Hierarchy

Parallelogram: Opp sides $||$, Opp sides $\cong$, Diagonals bisect.

Rectangle: 4 right angles. Diagonals $\cong$.

Rhombus: 4 equal sides. Diagonals $\perp$.

Square: Rectangle AND Rhombus.

Trapezoid: Exactly 1 pair parallel sides.

Kite: Adj sides $\cong$, Diagonals $\perp$.

Circles

Circumference: $C = 2\pi r = \pi d$

Area: $A = \pi r^2$

Arc Length: $L = 2\pi r (\frac{\theta}{360})$

Sector Area: $A = \pi r^2 (\frac{\theta}{360})$

Angles

Central: Angle = Arc

Inscribed: Angle = 1/2 Arc

3D Solids (Volume & Surface Area)

Prism/Cylinder:

$V = \text{BaseArea} \cdot h$

$SA = 2B + Ph$

Pyramid/Cone:

$V = \frac{1}{3}Bh$

$SA = B + \frac{1}{2}Pl$ ($l=$ slant height)

Sphere:

$V = \frac{4}{3}\pi r^3$

$SA = 4\pi r^2$

GEOMETRY Complete Reference