Back to Home
Geometry and Trigonometry

Geometry and Trigonometry

Introduction to geometry and trigonometry, exploring shapes, angles, and relationships forming the foundation for spatial reasoning and advanced mathematical problem-solving.

Geometry and Trigonometry

Introduction to geometry and trigonometry, exploring shapes, angles, and relationships forming the foundation for spatial reasoning and advanced mathematical problem-solving.

Linear Equations and Inequalities

4 min read

Concepts of Geometry

4 min read

Angle Relationships

3 min read

Congruence Proof

0:21

Congruence Proof Practice

3 tasks

Points, Lines, Planes

23 min read

Concepts of Geometry

4 min read
Mar 31, 2026
Free

In this lesson

1. Introduction to Geometry2. Undefined TermsKey idea3. Points, Lines, and IncidenceLine through two pointsNotation4. Segments and DistanceDistance5. Rays and DirectionProperties6. Collinearity and BetweennessCollinear pointsBetweenness7. AnglesTypes of angles8. Geometric Structure Summary
You will learn
  • Fundamental undefined terms in geometry
  • Relationships between points, lines, and planes
  • Basic geometric objects: segments, rays, angles
  • Core properties: distance, betweenness, collinearity
  • Axiomatic structure of Euclidean geometry

1. Introduction to GeometryLink to 1-introduction-to-geometry

Geometry studies shapes, positions, sizes, and spatial relationships in space.

It is constructed as a logical system based on:

  • Undefined terms (primitive concepts)
  • Axioms (assumed truths)
  • Theorems (derived results)
Note

Geometry is deductive: every statement must be justified from definitions or axioms.

2. Undefined TermsLink to 2-undefined-terms

Geometry is built on three fundamental undefined concepts:

ConceptIntuitionProperties
PointExact position in spaceNo length, width, or height
LineStraight infinite set of pointsInfinite length, no thickness
PlaneFlat 2D surfaceInfinite area, no thickness

Key ideaLink to 2-undefined-terms-key-idea

  • A point defines location only.
  • A line is determined by at least two points.
  • A plane contains infinitely many lines and points.

IMAGE: Diagram showing a point A, a line through A and B, and a plane extending infinitely

3. Points, Lines, and IncidenceLink to 3-points-lines-and-incidence

A fundamental relation in geometry is incidence (which objects lie on which).

Line through two pointsLink to 3-points-lines-and-incidence-line-through-two-points

A≠B⇒∃! AB↔A \neq B \Rightarrow \exists ! \ \overleftrightarrow{AB}A=B⇒∃! AB
Remember

Through any two distinct points, exactly one straight line exists.

NotationLink to 3-points-lines-and-incidence-notation

  • Line: AB↔\overleftrightarrow{AB}AB
  • Segment: AB‾\overline{AB}AB
  • Ray: AB→\overrightarrow{AB}AB

4. Segments and DistanceLink to 4-segments-and-distance

A line segment is a part of a line bounded by two endpoints.

AB‾={points between A and B}\overline{AB} = \{ \text{points between } A \text{ and } B \}AB={points between A and B}

DistanceLink to 4-segments-and-distance-distance

The distance between points AAA and BBB is the length of AB‾\overline{AB}AB:

AB=∣xB−xA∣(on a number line)AB = |x_B - x_A| \quad \text{(on a number line)}AB=∣xB​−xA​∣(on a number line)
Note

Distance is always non-negative: AB≥0AB \ge 0AB≥0

5. Rays and DirectionLink to 5-rays-and-direction

A ray has a starting point and extends infinitely in one direction.

AB→=ray starting at A passing through B\overrightarrow{AB} = \text{ray starting at } A \text{ passing through } BAB=ray starting at A passing through B

PropertiesLink to 5-rays-and-direction-properties

  • Has one endpoint
  • Infinite in one direction
  • Defines direction in geometry

6. Collinearity and BetweennessLink to 6-collinearity-and-betweenness

Collinear pointsLink to 6-collinearity-and-betweenness-collinear-points

Points are collinear if they lie on the same line.

A,B,C are collinear ⇔∃AB↔∋CA, B, C \text{ are collinear } \Leftrightarrow \exists \overleftrightarrow{AB} \ni CA,B,C are collinear ⇔∃AB∋C

BetweennessLink to 6-collinearity-and-betweenness-betweenness

A point BBB is between AAA and CCC if:

  • All three are collinear
  • AB+BC=ACAB + BC = ACAB+BC=AC
Important

Betweenness is a fundamental ordering concept in geometry.

7. AnglesLink to 7-angles

An angle is formed by two rays sharing a common endpoint.

∠ABC\angle ABC∠ABC
  • Vertex: BBB
  • Sides: BA→,BC→\overrightarrow{BA}, \overrightarrow{BC}BA,BC

Types of anglesLink to 7-angles-types-of-angles

TypeMeasure
Acute0∘<θ<90∘0^\circ < \theta < 90^\circ0∘<θ<90∘
Rightθ=90∘\theta = 90^\circθ=90∘
Obtuse90∘<θ<180∘90^\circ < \theta < 180^\circ90∘<θ<180∘
types of angles

8. Geometric Structure SummaryLink to 8-geometric-structure-summary

Geometry is built in layers:

  1. Undefined terms (point, line, plane)
  2. Relations (incidence, betweenness)
  3. Derived objects (segments, rays, angles)
  4. Theorems and proofs
Core idea

All geometric knowledge is derived from a small set of primitives using logical deduction.

LogoMath Course Online

Transforming math education
with interactive, engaging courses.

Explore

CoursesFAQ

Account

Sign InSign Up

Connect

GitHubGitHubContact

© 2026 Math Course Online. All rights reserved.