Exhaustive Study Guide for University Geometry: From Euclidean Basics to 3D Solids

Foundations of Euclidean Geometry and the Necessity of Undefined Terms

In Euclidean geometry, the study of geometric objects begins with the recognition that not all terms can be formally defined. Many words used in daily language are difficult to define without falling into circular reasoning. For example, if a straight line is defined as a line that lacks curves, and a curve is defined as a line that is not straight, no true understanding is achieved. Similarly, defining a line as a set of points extending in one direction requires a prior definition of the word direction. To avoid this, geometry acknowledges certain undefined terms—specifically the point, the line, and the plane—which are considered so elemental that their meanings are accepted as common knowledge. Any effort to describe them relies on synonyms or intuitive descriptions rather than formal mathematical definitions.

Mathematical truths that are accepted as evident without the need for demonstration are called postulates or axioms. Several fundamental axioms establish the behavior of geometric elements. For instance, it is a postulate that through any two distinct points, exactly one unique straight line can be drawn. Regarding perpendicularity, for any point located outside a specific line, there exists only one perpendicular line that can be drawn to it. Similarly, for a point located on the line itself, only one perpendicular can be constructed at that point. In terms of parallelism, through a point outside a given line, one and only one parallel line can be drawn. Furthermore, it is axiomatically accepted that two lines perpendicular to the same line are parallel to each other, and two lines parallel to the same line are likewise parallel to one another.

Fundamental Geometric Definitions and the Language of Angles

To ensure a rigorous study of geometry, specific notation and nomenclature must be followed. Points are always denoted using capital letters from the alphabet. The concept of "being between" is formally described such that for three distinct points $A$, $B$, and $C$, we say $B$ is between $A$ and $C$ if they are collinear and satisfy the condition that the segment length $AB$ plus $BC$ equals the total length $AC$. A segment (or trace) consists of the set of points on a line formed by two endpoints $A$ and $B$ and all points lying between them. Every segment possesses a unique midpoint $M$ such that $M$ belongs to the segment and the distance $AM$ equals the distance $MB$. A ray is formed when a point $P$ on a line divides it into two semi-lines; the ray consists of one of these semi-lines combined with the point of division $P$, which serves as the origin.

An angle is determined by the union of two rays that share a common origin called the vertex. If the rays are $BA$ and $BC$, the angle is denoted as $\angle ABC$. Geometrically, it can be defined as the set of points forming the two rays or as the measure of the opening between them. Angles are measured in degrees using the sexagesimal system. They are classified according to their measure: a right angle measures exactly $90^{\circ}$, an acute angle measures between $0^{\circ}$ and $90^{\circ}$ (exclusive), an obtuse angle measures between $90^{\circ}$ and $180^{\circ}$ (exclusive), and a straight (extended) angle measures exactly $180^{\circ}$. Adjacent angles are those that share a vertex and a common side, particularly when a ray originates from a point on a line. If two angles together form a straight angle, they are called supplementary angles because their measures sum to $180^{\circ}$. Conversely, if their sum is $90^{\circ}$, they are called complementary angles.

Relationships Between Lines and Transversals

Perpendicularity occurs when two lines intersect to form four congruent right angles, denoted by the symbol $\perp$. This definition extends to rays and segments if the lines containing them are perpendicular. Parallelism, denoted by //, describes two lines in the same plane that do not intersect and have no points in common. Lines that are not parallel and have an intersection point are termed secant lines. The distance between two distinct points $A$ and $B$ is a positive number representing the length of the unique path along the line connecting them, denoted simply as $AB$. Surface is defined as the set of points forming a sector of a plane divided by a geometric object.

When two distinct lines $L_1$ and $L_2$ are intersected by a third line $T$ at different points, the line $T$ is called a transversal. This intersection creates several pairs of angles with specific properties. Vertical (opposite) angles share a vertex and are congruent. Alternate internal angles are pairs on opposite sides of the transversal between the two lines, while alternate external angles are on opposite sides of the transversal outside the lines. Corresponding angles occupy the same relative position at each intersection. According to the fundamental theorems of geometry, a pair of alternate internal angles or corresponding angles are congruent if and only if the lines $L_1$ and $L_2$ are parallel. If the lines are parallel, the interior angles on the same side of the transversal are supplementary (summing to $180^{\circ}$).

Polygons: Definitions, Elements, and Classification

A polygon is a flat figure consisting of the union of a finite number of segments that satisfy specific conditions: segments only intersect at their endpoints, exactly two segments meet at each endpoint, and the entire figure can be drawn without lifting the pencil. The points where segments meet are called vertices, and the segments themselves are the sides. Diagonals are defined as segments connecting two non-consecutive vertices. The number of diagonals that can be drawn from a single vertex in a polygon with $n$ sides is given by the formula $n - 3$. The sum of the interior angles of any polygon is determined by the number of sides, following the general expression $(n - 2) \times 180^{\circ}$.

Polygons are classified using several criteria. Based on convexity, a polygon is convex if all its interior angles measure less than $180^{\circ}$; if any interior angle exceeds $180^{\circ}$, the polygon is concave. Based on regularity, a regular polygon is both equilateral (all sides congruent) and equiangular (all angles congruent). Examples include the equilateral triangle and the square. Polygons are also named by their number of sides: triangle ($3$), quadrilateral ($4$), pentagon ($5$), hexagon ($6$), heptagon ($7$), octagon ($8$), nonagon/eneagon ($9$), decagon ($10$), endecagon ($11$), and dodecagon ($12$). Generally, polygons with more than $10$ sides are often referred to simply by their number of sides.

Triangles and Their Secondary Elements

A triangle is a three-sided polygon formed by the intersection of three secant lines. The existence of a triangle is governed by the triangle inequality, which states that the measure of any side must be less than the sum of the other two sides. Triangles are classified by their sides—escalene (no equal sides), isosceles (at least two equal sides), and equilateral (all sides equal)—and by their angles—acutangle (all acute angles), rectangle (one right angle), and obtusangle (one obtuse angle). In any triangle, the sum of the interior angles is always $180^{\circ}$, and the measure of an exterior angle is equal to the sum of the two non-adjacent interior angles.

Secondary elements provide deeper structural insight into triangles. Altitudes are perpendicular segments from a vertex to the opposite side (or its extension), intersecting at the orthocenter. Angle bisectors divide internal angles into two equal parts and meet at the incenter, which is equidistant from the sides. Medians (transversals of gravity) connect a vertex to the midpoint of the opposite side, intersecting at the barycenter or center of gravity; the barycenter divides each median in a $1:2$ ratio. In a right triangle, the median to the hypotenuse is exactly half the length of the hypotenuse. Mediatrices (simetrales) are perpendicular lines at the midpoint of each side, meeting at the circumcenter, which is equidistant from the vertices. Midsegments (parallel lines) connect the midpoints of two sides; they are parallel to the third side and measure exactly half its length. Area is calculated as $A = \frac{base \times height}{2}$, or via Heron's formula: $A = \sqrt{s(s-a)(s-b)(s-c)}$, where the semiperimeter $s = \frac{a+b+c}{2}$.

Quadrilaterals and Circular Figures

Quadrilaterals are four-sided polygons classified by the parallelism of their sides. Parallelograms have two pairs of parallel sides and include squares, rectangles, rhombuses, and rhomboids. Their properties include congruent opposite sides and angles, supplementary consecutive angles, and diagonals that bisect each other (dimidiation). Trapezoids have exactly one pair of parallel sides (bases). The median of a trapezoid connects the midpoints of non-parallel sides and its length is the average of the bases: $MN = \frac{AB + DC}{2}$. The area is the product of the median and the height. Trapeziums have no parallel sides.

Circumference is the set of all points in a plane equidistant from a fixed center point, whereas a circle includes the circumference and its interior. Key elements include the radius ($r$), diameter ($d = 2r$), chord (segment joining two points on the circle), and arc (portion of the circumference). The perimeter of a circle is $P = 2\pi r$, and the area is $A = \pi r^2$. Special circular figures include the semicircle, circular sector (a "slice" determined by a central angle), circular segment, and the annulus (corona circular). The area of a circular sector is proportional to its central angle $\alpha$: $A_{sector} = \frac{\alpha}{360} \times \pi r^2$.

Geometry in Space and Solid Volumes

Three-dimensional geometry involves objects in space. Lines in space can be parallel (coplanar and non-intersecting), secant (intersecting), or skewed (alabeadas), which are non-coplanar and non-intersecting. A plane and a line can be parallel, the line can be contained in the plane, or they can be secant at a single point. Solids are sets of points enclosed by surfaces. Polyhedra are solids bounded by polygons (faces), with edges as intersections of faces and vertices as intersection points of edges. Euler's relationship for convex polyhedra states: $C + V = A + 2$, where $C$ is the number of faces, $V$ is the number of vertices, and $A$ is the number of edges.

Prisms have two congruent parallel bases and parallelogram lateral faces. Pyramids have a base and triangular lateral faces meeting at a common apex. The volume of any prism is $V = A_{base} \times h$. By contrast, the volume of a pyramid is exactly one-third that of a prism with the same base and height: $V = \frac{1}{3} \times A_{base} \times h$. Solids of revolution are generated by rotating a plane figure around an axis. A cylinder has volume $V = \pi r^2 h$ and surface area $A = 2\pi rh + 2\pi r^2$. A cone's volume is $V = \frac{1}{3} \pi r^2 h$ and its surface area is $A = \pi rg + \pi r^2$, where $g$ is the slant height (generatriz). A sphere, with all points equidistant from a center in 3D space, has volume $V = \frac{4}{3} \pi r^3$ and surface area $A = 4\pi r^2$.

Isometric Transformations and Similarity

Geometric transformations describe changes to a figure. Isometries are transformations that preserve distance, meaning size and shape remain constant while orientation or position may change. Rotations involve turning a figure by an angle $\alpha$ around a center $O$. In the Cartesian plane, rotating a point $P(x, y)$ around the origin by $90^{\circ}$ yields $(-y, x)$, by $180^{\circ}$ yields $(-x, -y)$, and by $270^{\circ}$ yields $(y, -x)$. Translations shift every point of a figure according to a translation vector $T(a, b)$, moving it a specific direction and distance. Reflections (symmetry) involve mirroring a figure across an axis (axial symmetry) or a point (central symmetry). For axial symmetry, each point and its image are equidistant from the axis.

Similarity (semejanza) occurs when figures have identical shapes but potentially different sizes; their angles are congruent and their corresponding sides are proportional. Triangles are similar if they meet criteria such as AA (two matching angles), LAL (two proportional sides and equal included angle), or LLL (all sides proportional). Homothety is a specific similarity transformation from a center $O$ with a ratio $k$. If $k > 1$, the figure is enlarged; if $0 < k < 1$, it is reduced; and if $k < 0$, the figure is inverted and placed on the opposite side of the center. The relationship is expressed as $k = \frac{OA'}{OA}$, where $A'$ is the image of point $A$.

Questions and Discussion

The material includes several self-reflection points regarding the purpose of teaching geometry. It models physical space and develops the capacity for abstraction, transitioning from empirical observation (measuring a triangle's angles) to deductive reasoning (using parallel lines and transversals to prove the sum is $180^{\circ}$). Discussion questions provided in the text ask students to consider if the center of a circle is a point of the circle itself (no, it is not on the circumference) and how many diameters pass through a specific point on the circumference (exactly one). Another hypothetical scenario asks Jorge to calculate the depth of a well using similar triangles formed by his line of sight from a height of $1.70\,m$ standing $0.9\,m$ from the edge. Activity prompts also include construction tasks using a compass and ruler to bisect angles or find the mediatrix of a segment, as well as using software like GeoGebra for dynamic geometric manipulation.