Chapter 1 · Tools of Geometry

Modeling and Defining with Undefined Terms

Use points, lines, and planes to model real objects, then use them to build precise definitions that work in both directions.

18 min read

You can already name points, lines, and planes. Now you will see why geometry starts with them and put them to work.

Every definition uses other words, and those words need meanings too. If you tried to define every word, you would end up going in circles. So geometry starts with three undefined terms, words you describe and use without defining: point, line, and plane.

You will use these terms in two ways. First, you will describe real objects with them. Then you will build precise definitions from them.

A model is a geometric figure that stands for a real object. It keeps the features that matter for your question and ignores the rest.

To choose a model, compare the object with these descriptions:

A point has a location but no size.
A line is straight, has no thickness, and extends without end in two opposite directions.
A plane is flat, has no thickness, and extends without end in all directions.

Real objects have size, thickness, and ends, so a model is always a simplification. An object with ends or edges can model only part of a line or a plane.

Three real objects and their models

Real object	Model	What it keeps	What it ignores
A town on a map of a whole country	A point	Its location	Its area and streets
A tight string between two nails	Part of a line	Its straightness	Its thickness and material
A flat tabletop	Part of a plane	Its flatness	Its thickness and material

The best model depends on your question. If you only need to show where a town is on a country map, a point works. If you are planning that town's bus routes, its area and streets matter, so a single point no longer works.

Worked example

Model the parts of a room

You want to describe how the parts of a rectangular room fit together, so thickness does not matter. Model each part with a point, a line, or a plane: the floor, the edge where one wall meets the floor, and the corner where two walls meet the floor.

Model the floor. It is a flat surface, and its thickness does not matter here, so use part of a plane. It is only part of a plane because the real floor stops at the walls.
Model the edge. It is straight, and its width does not matter here, so use part of a line. It is only part of a line because the real edge stops at the corners.
Model the corner. Only its location matters here, so use a point. You ignore any size the real corner has.
Check that the models fit. A postulate is a statement accepted without proof. One of the point-line-plane postulates says that if two different planes intersect, meaning they share at least one point, then their intersection is a line. Model the wall with a plane too. Then the wall and floor planes meet in a line, which agrees with your model of the edge.

Answer

The floor is modeled by part of a plane, the edge by part of a line, and the corner by a point.

Worked example

Use a model to explain a wobble

A three-legged stool and a four-legged table stand on a flat floor. It is given that the stool's three leg tips do not lie on one line. It is also given that one table leg is too short, so the table's four leg tips do not all lie in one plane. Use a model to explain why all three stool tips can touch the floor at once, but all four table tips cannot.

Choose the models. Use a plane for the floor, because only its flatness matters. Use a point for each leg tip, because only its location matters.
Reason about the stool. One of the point-line-plane postulates says that through any three noncollinear points, meaning points not on one line, there is exactly one plane. The three tips are given as noncollinear, so exactly one plane contains them. When that plane lies along the floor, all three tips touch.
Reason about the table. The four tips do not all lie in one plane, so they are not coplanar by the definition of coplanar. The floor is modeled by one plane, so it cannot contain all four tips at once. At least one tip stays off the floor, so the table can rock.
State the limit of this reason. The table rocks because its tips are not coplanar, not simply because it has four legs. If a table's four tips are coplanar, the plane that contains them can lie along this floor, so all four can touch.

Answer

All three stool tips can touch the floor because three noncollinear points lie in exactly one plane. All four table tips cannot touch it, because no single plane contains all four.

Now turn to definitions. A definition states the exact conditions that make a term apply. It works in both directions: the term gives you the conditions, and the conditions give you the term.

You already know that points are collinear when they lie on one line. Read that definition both ways:

If points are collinear, then they lie on one line.
If points lie on one line, then they are collinear.

Both statements are true. When you use a definition as a reason, check which direction you need.

Key concept

Undefined terms do two jobs

You do not define point, line, or plane. You use them to model real objects, keeping only what your question needs. You also use them to build precise definitions. A precise definition uses only undefined terms and ideas already established, states every condition, and works in both directions.

Some definitions need the order of points along a line, and some need distance. Both come from the Ruler Postulate, which is accepted without proof. It says that the points of a line can be matched one-to-one with the real numbers. A point's number is its coordinate. The distance between two points is the absolute value of the difference of their coordinates. So if $A$ has coordinate $a$ and $B$ has coordinate $b$ , the distance $AB$ is:

AB = |a - b|

Coordinates make the word between precise. Suppose $A$ , $B$ , and $C$ have coordinates $a$ , $b$ , and $c$ . Point $B$ is between $A$ and $C$ when the three points lie on one line and either $a < b < c$ or $c < b < a$ .

Now you can build other definitions, each from undefined terms and ideas that come before it. In the table, $A$ and $B$ are different points. Line $AB$ means the one line through them, because one of the point-line-plane postulates says that through any two different points there is exactly one line.

Definitions built from the undefined terms

Term	Precise definition	Built from
Line segment $\overline{AB}$	Points $A$ and $B$ , called endpoints, and all points on line $AB$ between $A$ and $B$	point, line, between
Ray $\overrightarrow{AB}$	All points of $\overline{AB}$ and all points $P$ on line $AB$ with $B$ between $A$ and $P$ ; $A$ is its endpoint	line segment, point, line, between
Angle	Two rays with the same endpoint, called the vertex, that do not lie on one line	ray, point, line
Right angle	An angle with measure exactly $90^\circ$ . Measures come from the Protractor Postulate, accepted without proof.	angle, measure
Parallel lines, $\ell \parallel m$	Two lines that are coplanar, meaning one plane contains both, and do not intersect	line, plane
Perpendicular lines, $\ell \perp m$	Two lines that intersect to form a right angle	line, right angle
Circle	All points in a plane at a given positive distance, the radius, from a given point in that plane, the center	point, plane, distance

Worked example

Use the definitions of segment and ray

By the Ruler Postulate, five points on one line have these coordinates: $A$ at $2$ , $B$ at $5$ , $C$ at $9$ , $D$ at $11$ , and $E$ at $0$ . Decide which of $B$ , $D$ , and $E$ lie on $\overline{AC}$ and which lie on $\overrightarrow{AC}$ .

Test point B. Since 5 is between 2 and 9, B is between A and C by the definition of between. The conditions give the term here, so B is on segment AC by the definition of line segment. Ray AC contains all points of segment AC by the definition of ray, so B is on the ray too.
$2 < 5 < 9$
Test point D on the segment. Here the term gives the conditions: by the definition of line segment, every point of segment AC is an endpoint or is between A and C. D is not an endpoint, and 11 is greater than both 2 and 9, so D is not between A and C. So D is not on segment AC.
$11 > 9 > 2$
Test point D on the ray. Since 9 is between 2 and 11, C is between A and D. By the definition of ray, D is on ray AC.
$2 < 9 < 11$
Test point E. It is not an endpoint, and 0 is less than both 2 and 9, so E is not between A and C and is not on the segment. Also, 9 is not between 2 and 0, so C is not between A and E. By the definition of ray, E is not on the ray.
$0 < 2 < 9$

Answer

Point $B$ lies on both $\overline{AC}$ and $\overrightarrow{AC}$ , point $D$ lies only on $\overrightarrow{AC}$ , and point $E$ lies on neither, although all three lie on line $AC$ .

Worked example

Test a shortened definition

A classmate drops the words about a plane and says: "A circle with center $O$ and radius $5$ is all points at distance $5$ from $O$ ." Show that this version fails. Then use the full definition to decide whether $P$ and $Q$ are on the circle. Both lie in the circle's plane on a line through $O$ , with coordinates $O$ at $3$ , $P$ at $8$ , and $Q$ at $6$ .

Test the direction from conditions to term. Model the outer surface of a ball with center O and radius 5 as all points in space at distance 5 from O. This surface meets your classmate's conditions. It does not lie in a plane, so it is not a circle. The shortened version fails.
Test P with the Ruler Postulate. P is in the circle's plane and its distance from O is 5, so the conditions give the term: P is on the circle by the definition of circle.
$OP = |8 - 3| = |5| = 5$
Test Q the same way. The term gives the conditions: every point on the circle is at distance 5 from O. The distance from O to Q is 3, not 5, so Q is not on the circle.
$OQ = |6 - 3| = |3| = 3 \neq 5$

Answer

The shortened version fails because a ball's surface meets its conditions but is not a circle. By the full definition, $P$ is on the circle and $Q$ is not.

Point, line, and plane are where geometry starts. Use them to model real objects by keeping only what your question needs. Use them to write definitions in which every word does a job and both directions are true.

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