Comprehensive Guide to Geometry Axioms, Theorems, and Formulas

Axioms of Inequality and Algebraic Relationships

The principles of inequality define how relative values change or remain consistent under specific operations. If $b > a$, then it follows that $c + b > c + a$, $c - b > c - a$, and $c \times b > c \times a$. Conversely, if $b < a$, then $d + b < c + a$. In cases where an addition results in a sum, such as if $c + b = a$, it can be concluded that $c < a$ and $b < a$ is false, rather the transcript states $c > a$ and $b > a$ is not possible if they are parts of the whole, implying $a$ is the larger sum; specifically, the rule notes if $c + b = a$, then $c < a$ and $b < a$ (assuming positive values). Additional logical deductions for angles include: if $m(\angle ABD) > m(\angle XYL)$ and $m(\angle DBC) < m(\angle LYZ)$, various comparisons can be made regarding the composite angles.

Exterior Angles and Triangle Inequalities

The Exterior Angle Theorem states that the measure of an exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent interior angles. Formally, for a triangle $ABC$ with an exterior angle at $C$ labeled as $ACD$, the relationship is $m(\angle ACD) = m(\angle A) + m(\angle B)$. Because the measure of the exterior angle is the sum of these two interior angles, it inherently follows that the exterior angle is greater than either individual interior angle: $m(\angle ACD) > m(\angle A)$ and $m(\angle ACD) > m(\angle B)$. Furthermore, in any triangle, the side opposite a greater angle is longer than the side opposite a smaller angle. This is expressed in Rule 1: If $AC > AB$, then $m(\angle B) > m(\angle C)$. Conversely, if $m(\angle B) > m(\angle C)$, then $AC > AB$. A specific note for a single triangle emphasizes that the greatest side is always opposite the greatest angle; for instance, if $B$ is the greatest angle (such as an obtuse angle), then $AC$ is the longest side, leading to the inequalities $AB < AC$ and $BC < AC$.

The Triangle Inequality Theorem

The triangle inequality theorem establishes the necessary relationship between the lengths of the three sides of a triangle ($AB$, $BC$, and $AC$). It states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. This can be summarized by Ruler 2: $|BC - AC| < AB < BC + AC$. This inequality shows that the length of side $AB$ must be strictly greater than the absolute difference of the other two sides and strictly less than their sum. If the sum of two sides is equal to the third side ($AC = BC + AB$), then the points $A$, $B$, and $C$ are collinear. If the sum of two sides is less than the third side ($AC > BC + AB$), the points $A$, $B$, and $C$ are considered vertices of a triangle only if the inequality holds; however, specifically, the notes state if $AC > BC + AB$, they are vertices.

Geometric Projections and Theorems of Pythagoras and Euclid

Projections describe the mapping of points or segments onto a line. The projection of a point $C$ onto a line $L$ is a point, and the length of the projection of a segment is always less than or equal to the length of the segment itself ($\text{length of projection} \le \text{length of segment}$). If a segment is perpendicular to the line ($AB \perp L$), the length of the projection is $0$. If a segment is parallel to the line ($AB \parallel L$), then the length of the projection is equal to the original segment ($A'B' = AB$).

In right-angled triangle calculations, Pythagoras' theorem provides the relations: $(AC)^2 = (AB)^2 + (BC)^2$, $(AB)^2 = (AC)^2 - (BC)^2$, and $(BC)^2 = (AC)^2 - (AB)^2$. Euclid's theorems provide additional relations for a right triangle with an altitude $AD$ dropped from the right angle $A$ to the hypotenuse $BC$:

1. $(AB)^2 = BC \times BD$
2. $(AC)^2 = CB \times CD$
3. $(AD)^2 = DC \times DB$
4. $AD = \frac{AC \times AB}{BC}$

Circle Geometry and Coordinate Calculations

A circle is defined by its center $M$ and radius $r$ (segments such as $MA$ or $MB$). The diameter $d$ is defined as $d = 2r$. The circumference of a circle is calculated as $C = 2\pi r$ or $C = \pi d$. The area of a circle is given by $Area = \pi r^2$, where $\pi$ is approximately $\frac{22}{7}$ or $3.14$.

In coordinate geometry, the location of a point $(x, y)$ relative to the axes is determined by absolute values: the distance to the x-axis is $|y|$ (e.g., for point $(4, -3)$, the distance is $|-3| = 3$ units), and the distance to the y-axis is $|x|$. The distance $d$ between two points $A(x_1, y_1)$ and $B(x_2, y_2)$ is calculated using the formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$. The distance from a point $A(x, y)$ to the origin $B(0, 0)$ is $d = \sqrt{x^2 + y^2}$. The slope $m$ of a line passing through these points is $m = \frac{y_2 - y_1}{x_2 - x_1}$. If a line is parallel to the x-axis, its slope is $0$. If a line is parallel to the y-axis, its slope is undefined. For points to be collinear, the slope of $AB$ must equal the slope of $BC$. To prove points $A$, $B$, and $C$ lie on the same circle with center $M$, one must prove $MA = MB = MC$.

Classification of Triangles by Sides and Angles

Triangles are classified by side lengths: an equilateral triangle has $AB = BC = AC$, and an isosceles triangle has at least two equal sides (e.g., $AB = AC$). To determine if a triangle is acute, right, or obtuse at vertex $B$, compare the square of the longest side (assume $AC$) to the sum of the squares of the other two sides:

1. If $(AC)^2 < (AB)^2 + (BC)^2$, the triangle is acute.
2. If $(AC)^2 = (AB)^2 + (BC)^2$, the triangle is right-angled at $B$.
3. If $(AC)^2 > (AB)^2 + (BC)^2$, the triangle is obtuse at $B$.

General area and perimeter formulas for triangles include: $Area = \frac{1}{2} \times \text{base} \times \text{height}$ and $Perimeter = \text{Sum of all sides}$. For a square with side $S$, $Area = S^2$ and $Perimeter = 4S$. For a rectangle with length $L$ and width $W$, $Area = L \times W$ and $Perimeter = 2 \times (L + W)$.

Properties and Volume of Geometric Solids

The volume and surface area of a Right Circular Cylinder are determined by the radius $r$ and height $h$:

• $Volume = \pi r^2 h$ (Base Area $\times$ Height)
• $Lateral Area = 2\pi rh$ (Base Circumference $\times$ Height)
• $Total Area = 2\pi rh + 2\pi r^2$ (Lateral Area + Double Base Area), which simplifies to $2\pi r(r + h)$.

For a Right Prism (including Triangular and Quadrilateral Prisms):

• $Volume = \text{base area} \times \text{height}$
• $Lateral Area = \text{base perimeter} \times \text{height}$
• $Total Area = \text{lateral area} + (2 \times \text{base area})$.

Identifying Specific Quadrilaterals

To prove a quadrilateral $ABCD$ is a specific shape, certain conditions must be met:

1. Parallelogram: Prove opposite sides are equal ($AD = BC$ and $CD = AB$).
2. Rhombus: Prove all sides are equal ($AD = CD = BC = AB$).
3. Rectangle: Prove it is a parallelogram ($AD = BC$ and $CD = AB$) and the diagonals are equal ($AC = BD$).
4. Square: Prove all sides are equal ($AD = CD = BC = AB$) and the diagonals are equal ($AC = BD$).