In this lesson:
By the end of this lesson, you should be able to:
Try to define the word "point" without using a picture.
Every language must have starting words it cannot define — mathematics calls these the undefined terms.
These four are accepted by intuition — not defined from simpler terms
"Undefined" does not mean "we have no idea what it is."
The correct interpretation:
Which of these are undefined terms in Euclidean geometry?
Think before advancing — identify each category.
Undefined terms (starting points, not built from simpler terms):
Defined terms (built from undefined notions):
A precise mathematical definition must be:
Example: "A circle is the set of all points in a plane equidistant from a given center point" captures circles exactly — nothing more, nothing less.
An angle is formed by two rays that share a common endpoint.
Key distinction: An angle uses rays, not lines.
True or false: "An angle is formed by two lines."
Think about the definition before advancing.
False. An angle is formed by two rays, not two lines.
Every point on this circle is the same distance from center
The circle and the disk are different objects:
In everyday language, "draw a circle" means the filled region. In mathematics, a circle is only the boundary curve.
A filled round shape — is it a circle? What's the precise difference?
Answer in one sentence using the definition before advancing.
No — a filled round shape is a disk, not a circle.
Watch out: Never include the interior when working with circle theorems — theorems about circles describe points on the boundary.
Every defined term traces back to the four undefined starting points.
✓ Geometry starts with four undefined terms: point, line, distance along a line, distance around a circular arc
✓ Defined terms are built from undefined ones: line segment, angle, circle all trace back to these
✓ Definitions are precise and reversible — they fully characterize their objects
Watch out: Definitions are not arbitrary descriptions — they are agreed-upon meanings that enable proof
Watch out: A circle is the boundary only — the filled region is a disk
Watch out: Angles are formed by rays (one endpoint each), not lines
In Deck 2, we define two more fundamental objects — and show how precision enables proof:
The five definitions you learn in both decks form the foundation for all of high school geometry.
Click to begin the narrated lesson
Define geometric terms precisely