4.1: Euclidean geometry

Last updated
Save as PDF

Page ID: 4887

$ \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } $ $ \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} $$\newcommand{\id}{\mathrm{id}}$ $ \newcommand{\Span}{\mathrm{span}}$ $ \newcommand{\kernel}{\mathrm{null}\,}$ $ \newcommand{\range}{\mathrm{range}\,}$ $ \newcommand{\RealPart}{\mathrm{Re}}$ $ \newcommand{\ImaginaryPart}{\mathrm{Im}}$ $ \newcommand{\Argument}{\mathrm{Arg}}$ $ \newcommand{\norm}[1]{\| #1 \|}$ $ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$ $ \newcommand{\Span}{\mathrm{span}}$ $\newcommand{\id}{\mathrm{id}}$ $ \newcommand{\Span}{\mathrm{span}}$ $ \newcommand{\kernel}{\mathrm{null}\,}$ $ \newcommand{\range}{\mathrm{range}\,}$ $ \newcommand{\RealPart}{\mathrm{Re}}$ $ \newcommand{\ImaginaryPart}{\mathrm{Im}}$ $ \newcommand{\Argument}{\mathrm{Arg}}$ $ \newcommand{\norm}[1]{\| #1 \|}$ $ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$ $ \newcommand{\Span}{\mathrm{span}}$$\newcommand{\AA}{\unicode[.8,0]{x212B}}$

Euclidean geometry, sometimes called parabolic geometry, is a geometry that follows a set of propositions that are based on Euclid's five postulates.

There are two types of Euclidean geometry: plane geometry, which is two-dimensional Euclidean geometry, and solid geometry, which is three-dimensional Euclidean geometry.

The most basic terms of geometry are a point, a line, and a plane. A point has no dimension (length or width), but it does have a location. A line is straight and extends infinitely in the opposite directions. A plane is a flat surface that extends indefinitely.

Points

Definitions:

Please refer to the image below for examples.

$alt$

Collinear points: points that lie on the same straight line or line segment. Points A, B, and C are collinear.

Line Segment: a straight line with two endpoints. Lines AC, EF, and GH are line segments

Ray: a part of a straight line that contains a specific point. Any of the below line segments could be considered a ray

Intersection point: the point where two straight lines intersect, or cross. Point I is the intersection point for lines EF and GH.

Midpoint: a point in the exact middle of a given straight line segment. Point B is the midpoint of line AC.

Think Out Loud

How many different lines can you draw through a fixed point? How many different lines can you draw through two fixed points? How many different lines can you draw through three fixed points?

Angles

Definitions:

Angle: $\measuredangle ACB$. Normally, Angle is measured in degrees ($^0$) or in radians rad).

Right angle: Angles which measure 90° - $\measuredangle ABC$

Obtuse angle: Angles which measure > 90° - $\measuredangle CDE$

Acute angle: Angles which measure < 90° - $\measuredangle FDE$

Straight angle: Angles which measure 180° $\measuredangle CDF$

Reflex angle: A reflex angle is an angle, that is measured > 180°, which adds to an angle to make 360° - $\measuredangle CDE$'s reflex angle is $\measuredangle CDF + \measuredangle FDE$

$alt$

Adjacent angles: Have the same vertex and share a side. $\measuredangle HRL, \, \measuredangle HRO$ are adjacent.

Complementary angles: add up to 90°. $\measuredangle PRQ, \, \measuredangle QRI$ are complementary angles.

Supplementary angles: add up to 180°. $\measuredangle JSN, \, \measuredangle NSK$ are supplementary angles.

Vertical angles (X property): Angles which share line segments and vertexes are equivalent. $\measuredangle JSR, \, \measuredangle OST$ are vertical angles. They share the same degree value.

Corresponding angles (F property): Angles which share a line segment that intersects with parallel lines, and are in the same relative position on each respective parallel line, are equivalent. $\measuredangle IRQ, \, \measuredangle KUQ$ are corresponding angles. They share the same degree value.

Alternate interior angles (Z property): Angles which share a line segment that intersects with parallel lines, and are in opposite relative positions on each respective parallel line, are equivalent. $\measuredangle HRS, \, \measuredangle RST$ are alternate interior angles. They share the same degree value.

Bisecting an Angle: To bisect an angle is to draw a line concurrent line through the angle's vertex which splits the angle exactly in half. This is possible using a compass and an unmarked straightedge.

Trisecting an Angle: To trisect an angle is to use the same procedure as bisecting an angle, but to use two lines and split the angle exactly in thirds. This is an ancient impossibility - it is impossible to accomplish using a compass and an unmarked straightedge.

Lines

Definitions:

Parallel lines: Lines which, drawn on a 2-dimensional plane, may extend forever in either direction without ever intersecting. Lines $HI$ and $JK$ are parallel.

Perpendicular lines: Lines which intersect at exactly a 90° angle. Lines $HI$ and $MP$ are perpendicular.

Concurrent lines: Lines that all intersect at the same point. Lines $HI, \, LQ, \, MP, \, NO$ are concurrent.

Skew lines: Lines which, drawn in a 3-dimensional space, are both neither parallel nor perpendicular and do not intersect.

Perpendicular Bisector: A line that is perpendicular to a given line and bisects it is called a perpendicular bisector.

Example $\PageIndex{1}$:

Draw a cube and connect all the edges. Can you find two skew lines?

Planes

Definition:

A plane is a two-dimensional space that extends infinitely in all directions. For example, the graph of functions takes place on a Cartesian plane or a plane with coordinates. The plane continues in both the x and, y directions.

Axioms

The points, lines, and planes are objects with the relations given by the following axioms:

There is a unique line passing through two distinct points.
If two points lie in a plane, then any plane containing those points lies in the plane.
There is a unique plane containing three non-collinear points.
If two planes meet, then their intersection is a line.

Euclid's Five Postulates

Euclid's five postulates can be stated as follows

It is possible to draw a straight line segment joining any two points.
It is possible to indefinitely extend any straight line segment continuously in a straight line.
Given any straight line segment, it is possible to draw a circle having the segment as a radius and one endpoint as its center.
All right angles are equal to each other or congruent
Through a given point not on a given straight line, only one line can be drawn parallel to a given line.