Comprehensive Geometry Properties and Formulas Reference Guide

Properties of Equality and Operations

The fundamental properties of equality provide the basis for algebraic manipulation and geometric proofs. The Addition Property of Equality states that if $a = b$, then $a + c = b + c$. Similarly, the Multiplication Property of Equality specifies that if $a = b$, then $a \cdot c = b \cdot c$, where the condition contextually implies $c \neq 0$ for maintaining equality in division. The Subtraction Property of Equality states that if $a = b$, then $a - c = b - c$, and the Division Property of Equality states that if $a = b$, then $\frac{a}{c} = \frac{b}{c}$.

The Reflexive Property of Equality establishes that any value is equal to itself, expressed as $a = a$. The Symmetric Property of Equality states that if $a = b$, then $b = a$. The Transitive Property of Equality is used to link three values: if $a = b$ and $b = c$, then $a = c$. Furthermore, the Substitution Property of Equality allows that if $a = b$, then $a$ can be substituted for $b$ (or vice versa) in any equation or expression. The Distributive Property covers both sums and differences, where $a(b + c) = ab + ac$ and $a(b - c) = ab - ac$.

Properties of Congruence and Parallel Lines

Geometric congruence applies to segments and angles through specific properties. The Reflexive Property of Congruence states that for any segment $AB$, $AB \cong AB$, and for any angle $A$, $\angle A \cong \angle A$. The Symmetric Property of Congruence states that if $AB \cong CD$, then $CD \cong AB$, and if $\angle A \cong \angle B$, then $\angle B \cong \angle A$. The Transitive Property of Congruence establishes that if $AB \cong CD$ and $CD \cong EF$, then $AB \cong EF$; similarly, if $\angle A \cong \angle B$ and $\angle B \cong \angle C$, then $\angle A \cong \angle C$.

In the context of parallel lines, the Transitive Property of Parallel Lines states that if line $p \parallel q$ and line $q \parallel r$, then line $p \parallel r$.

Triangle and Polygon Theorems

The Triangle Inequality Theorem states that for any triangle with sides $AB$, $BC$, and $AC$, the sum of any two sides must be greater than the third side: $AB + BC > AC$, $AC + BC > AB$, and $AB + AC > BC$. The Pythagorean Inequalities Theorem provides a method to classify triangles based on the relation of the square of the longest side ($c$) to the sum of the squares of the shorter sides ($a$ and $b$): if $c^2 < a^2 + b^2$, the triangle $\triangle ABC$ is acute; if $c^2 > a^2 + b^2$, the triangle $\triangle ABC$ is obtuse.

The Triangle Sum Theorem mandates that the sum of interior angles in a triangle is $180^\circ$, expressed as $m\angle A + m\angle B + m\angle C = 180^\circ$. The Exterior Angle Theorem states that the measure of an exterior angle of a triangle is equal to the sum of the measures of its two remote interior angles: $m\angle 1 = m\angle A + m\angle B$. The Triangle Midsegment Theorem states that if a segment connects the midpoints of two sides (e.g., segment $DE$ in $\triangle ABC$), then $DE \parallel AC$ and $DE = \frac{1}{2}AC$.

For polygons, the Polygon Interior Angles Theorem states that the sum of the interior angle measures of an $n$-gon is $(n - 2) \times 180^\circ$. The Polygon Exterior Angles Theorem states that the sum of the exterior angle measures (one at each vertex) of any convex polygon is $360^\circ$. The Trapezoid Midsegment Theorem defines the midsegment $MN$ of a trapezoid as being parallel to both bases ($MN \parallel AB$ and $MN \parallel DC$) with a length of $MN = \frac{1}{2}(AB + CD)$.

Coordinate Geometry Formulas

Coordinate geometry utilizes various formulas to define lines and shapes on a plane. The Slope ($m$) between two points $(x_1, y_1)$ and $(x_2, y_2)$ is calculated as $m = \frac{y_2 - y_1}{x_2 - x_1}$. Linear equations can be expressed in Standard Form ($Ax + By = C$), Slope-Intercept Form ($y = mx + b$), and Point-Slope Form ($y - y_1 = m(x - x_1)$).

The Midpoint Formula for the center between $(x_1, y_1)$ and $(x_2, y_2)$ is given by $\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$. The Distance Formula between those same two points is $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$. For a circle, the Standard Equation is $(x - h)^2 + (y - k)^2 = r^2$, where $(h, k)$ represents the center and $r$ represents the radius.

Geometric Mean Theorems

The Geometric Mean (Altitude) Theorem states that in a right triangle, the square of the altitude to the hypotenuse is equal to the product of the segments of the hypotenuse: $CD^2 = AD \cdot BD$.

The Geometric Mean (Leg) Theorem provides two relationships: $\frac{DB}{CB} = \frac{CB}{AB}$, which simplifies to $CB^2 = DB \cdot AB$, and $\frac{AD}{AC} = \frac{AC}{AB}$, which simplifies to $AC^2 = AD \cdot AB$. This theorem states that each leg of a right triangle is the geometric mean of the hypotenuse and the segment of the hypotenuse adjacent to that leg.

Right Triangles and Trigonometry

The Pythagorean Theorem states that in a right triangle, $a^2 + b^2 = c^2$. Special right triangles possess unique side ratios. In a $45^\circ-45^\circ-90^\circ$ triangle, the hypotenuse is the leg length ($n$) multiplied by $\sqrt{2}$, or $\text{hypotenuse} = \text{leg} \cdot \sqrt{2}$. In a $30^\circ-60^\circ-90^\circ$ triangle, the hypotenuse is twice the shorter leg ($2n$), and the longer leg is the shorter leg ($n$) multiplied by $\sqrt{3}$.

Trigonometric ratios for an angle $A$ are defined as $\sin(A) = \frac{BC}{AB}$ (opposite/hypotenuse), $\cos(A) = \frac{AC}{AB}$ (adjacent/hypotenuse), and $\tan(A) = \frac{BC}{AC}$ (opposite/adjacent). Inverse trigonometric functions are used to find angle measures: $\sin^{-1}\left(\frac{BC}{AB}\right) = m\angle A$, $\cos^{-1}\left(\frac{AC}{AB}\right) = m\angle A$, and $\tan^{-1}\left(\frac{BC}{AC}\right) = m\angle A$. Conversions between units specify that $180^\circ = \pi\,\text{radians}$.

The relationship between sine and cosine of complementary angles states that if $A$ and $B$ are complementary, then $\sin(A) = \cos(90^\circ - A) = \cos(B)$ and $\cos(A) = \sin(90^\circ - A) = \sin(B)$.

General Triangle Laws and Area

For any non-right triangle, the area can be found using $\text{Area} = \frac{1}{2}bc \sin(A)$, $\text{Area} = \frac{1}{2}ac \sin(B)$, or $\text{Area} = \frac{1}{2}ab \sin(C)$. The Law of Sines is expressed as $\frac{\sin(A)}{a} = \frac{\sin(B)}{b} = \frac{\sin(C)}{c}$ or alternately $\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}$. The Law of Cosines is used to solve for sides or angles: $a^2 = b^2 + c^2 - 2bc \cos(A)$, $b^2 = a^2 + c^2 - 2ac \cos(B)$, or $c^2 = a^2 + b^2 - 2ab \cos(C)$.

Circle Calculations and Relationships

Arc length for arc $AB$ is calculated as $\text{Arc length of AB} = \frac{mAB}{360^\circ} \cdot 2\pi r$. The area of a sector $APB$ is calculated as $\text{Area of sector APB} = \frac{mAB}{360^\circ} \cdot \pi r^2$. Central angles have a measure equal to their intercepted arc: $m\angle ACB = mAB$. Inscribed angles are half the measure of their intercepted arc: $m\angle ADB = \frac{1}{2}(mAB)$.

When a tangent and an intersected chord meet at a point on the circle, the angle measures are $\angle 1 = \frac{1}{2}(mAB)$ and $\angle 2 = \frac{1}{2}(mBCA)$. For relationships involving segments and angles:

1. Two chords: $\angle 1 = \frac{1}{2}(mAC + mDB)$ and segments satisfy $EA \cdot EB = EC \cdot ED$.
2. Two secants: $\angle 1 = \frac{1}{2}(mBD - mAC)$ and segments satisfy $EA \cdot EB = EC \cdot ED$.
3. Tangent and Secant: $\angle 1 = \frac{1}{2}(mAD - mAC)$ and segments satisfy $EA^2 = EC \cdot ED$.
4. Two tangents: $\angle 1 = \frac{1}{2}(mACB - mAB)$ and segments satisfy $EA = EB$.

Geometric Formulas for Perimeter, Area, and Volume

For 2D shapes:

• Square: $P = 4s$; $A = s^2$
• Rectangle: $P = 2l + 2w$; $A = lw$
• Triangle: $P = a + b + c$; $A = \frac{1}{2}bh$
• Parallelogram: $A = bh$
• Trapezoid: $A = \frac{1}{2}h(b_1 + b_2)$
• Circle: $C = \pi d$ or $C = 2\pi r$; $A = \pi r^2$
• Rhombus/Kite: $A = \frac{1}{2}d_1 d_2$
• Regular n-gon: $A = \frac{1}{2}ap$ or $A = \frac{1}{2}ans$ where $a$ is the apothem, $n$ is number of sides, and $s$ is side length.

For 3D shapes:

• Prism: Lateral area $L = Ph$; Surface area $S = 2B + L$; Volume $V = Bh$.
• Cylinder: Lateral area $L = 2\pi rh$; Surface area $S = 2B + L = 2\pi r^2 + 2\pi rh$; Volume $V = \pi r^2 h$.
• Pyramid: Lateral area $L = \frac{1}{2}Pl$; Surface area $S = B + L = B + \frac{1}{2}Pl$; Volume $V = \frac{1}{3}Bh$ (where $l$ is slant height).
• Cone: Lateral area $L = \pi rl$; Surface area $S = B + L = \pi r^2 + \pi rl$; Volume $V = \frac{1}{3}\pi r^2 h$.
• Sphere: Surface area $S = 4\pi r^2$; Volume $V = \frac{4}{3}\pi r^3$.