Foundations of Geometry

The Language Geometry Speaks

Over two thousand years ago, the Greek mathematician Euclid sat down and asked a remarkable question: what is the smallest set of assumptions from which all of geometry can be built? His answer — a handful of definitions, postulates, and common notions — launched a tradition of rigorous reasoning that still shapes mathematics today.

Before you can prove theorems or solve problems, you need to speak the language. This unit introduces the vocabulary and notation that every geometric argument depends on.

Undefined Terms: Where It All Begins

Geometry starts with three terms so fundamental they cannot be defined using simpler concepts. Instead, we describe them:

• Point: A location in space with no size — no length, width, or height. We name points with capital letters: point A, point B.
• Line: A straight path extending infinitely in both directions, with no thickness. A line through points A and B is written

  $$\overleftrightarrow{AB}$$

  .
• Plane: A flat surface extending infinitely in all directions, with no thickness. A plane is named by three non-collinear points or a single capital letter.

These three ideas — a location, a path, and a surface — generate everything else in Euclidean geometry.

Segments, Rays, and Distance

From points and lines, we build more specific objects:

• Line segment: The part of a line between two endpoints. Segment AB is written

  $$\overline{AB}$$

  , and its length is written

  $$AB$$

  (no bar).
• Ray: A part of a line that starts at one endpoint and extends infinitely in one direction. Ray AB (starting at A, through B) is written

  $$\overrightarrow{AB}$$

  .
• Opposite rays: Two rays that share an endpoint and together form a line.

Distance and the Ruler Postulate: The distance between two points on a number line is the absolute value of the difference of their coordinates. If point A is at coordinate 3 and point B is at coordinate 8:

$$AB=|8-3|=5$$

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The Segment Addition Postulate: If point B is between points A and C on a segment, then

$$AB+BC=AC$$

. This seemingly simple idea is the foundation for solving many distance problems.

The Midpoint: The midpoint of a segment is the point that divides it into two congruent halves. If M is the midpoint of

$$\overline{AB}$$

, then

$$AM=MB=\frac{1}{2}\cdot AB$$

. You can find the midpoint on a coordinate plane using the midpoint formula:

$$M=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$$

Example: Find the midpoint of a segment with endpoints at

$$(2,8)$$

and

$$(6,4)$$

.

$$M=\left(\frac{2+6}{2},\frac{8+4}{2}\right)=(4,6)$$

❓

Concept Check

Points P, Q, and R are collinear with Q between P and R. If PQ = 7 and PR = 19, what is QR?

▸

By the Segment Addition Postulate, PQ + QR = PR. Substituting: 7 + QR = 19, so QR = 12.

Angles and Their Measurement

An angle is formed by two rays that share a common endpoint called the vertex. The angle formed by rays BA and BC is written

$$\angle ABC$$

or

$$\angle B$$

(when there is no ambiguity).

Angles are measured in degrees, with a full rotation equal to 360 degrees.

Angle Classification:

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The Angle Addition Postulate: If ray BD is in the interior of

$$\angle ABC$$

, then

$$m\angle ABD+m\angle DBC=m\angle ABC$$

. This works just like the Segment Addition Postulate, but for angles.

Special Angle Pairs

When two angles share a relationship, we can use that relationship to find unknown measures.

Complementary angles: Two angles whose measures add to 90 degrees. If

$$m\angle A=35°$$

, its complement has measure

$$90°-35°=55°$$

.

Supplementary angles: Two angles whose measures add to 180 degrees. If

$$m\angle B=110°$$

, its supplement has measure

$$180°-110°=70°$$

.

Vertical angles: When two lines intersect, they form two pairs of vertical angles. Vertical angles are always congruent (equal in measure).

Linear pair: Two adjacent angles that form a straight line. The angles in a linear pair are always supplementary.

Solving with angle relationships: These relationships turn geometric diagrams into algebra problems. If you know angles are complementary, set their sum equal to 90. If supplementary, set their sum equal to 180. If vertical, set them equal.

Example: Two vertical angles have measures

$$(5x-12)°$$

and

$$(3x+24)°$$

. Since vertical angles are congruent:

$$5x-12=3x+24$$

$$2x=36$$

$$x=18$$

Each angle measures

$$5(18)-12=78°$$

.

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❓

Concept Check

Two angles form a linear pair. One angle measures (3x + 10) degrees and the other measures (2x + 20) degrees. Find x and the measure of each angle.

▸

Linear pair angles are supplementary, so they add to 180 degrees: (3x + 10) + (2x + 20) = 180. Combining: 5x + 30 = 180, so 5x = 150 and x = 30. The angles measure 3(30) + 10 = 100 degrees and 2(30) + 20 = 80 degrees.

Basic Constructions

A construction uses only a compass and a straightedge — no rulers or protractors. This restriction forces you to rely on geometric properties rather than measurement.

Construction 1: Copying a segment. Given

$$\overline{AB}$$

, you can create a congruent segment starting at any point. Place the compass point on A, open to B, then transfer that width to the new location.

Construction 2: Bisecting a segment. The perpendicular bisector of a segment passes through the midpoint and is perpendicular to the segment. To construct it, draw arcs of equal radius from each endpoint; the line through the two intersection points is the perpendicular bisector.

Construction 3: Bisecting an angle. The angle bisector divides an angle into two congruent angles. From the vertex, draw an arc crossing both rays. From each intersection, draw equal arcs. The ray from the vertex through the new intersection point is the bisector.

Construction 4: Copying an angle. You can replicate any angle at a new location. Draw an arc from the original vertex, intersecting both rays. At the new vertex, draw the same arc. Then transfer the chord length between the original intersection points to the new arc, fixing the second ray.

Each construction relies on a geometric principle — congruent circles produce congruent arcs, and congruent arcs subtend congruent chords. The compass and straightedge translate these principles into physical marks on paper.

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❓

Concept Check

You construct the perpendicular bisector of a 10-centimeter segment. What two things are true about the point where the bisector crosses the segment?

▸

The bisector crosses the segment at its midpoint, which is 5 centimeters from each endpoint. At that point, the bisector is perpendicular to the segment, forming a 90-degree angle.

Collinear and Coplanar Points

Three or more points are collinear if they all lie on the same line. Points that do not lie on the same line are called non-collinear. It takes at least three non-collinear points to define a unique plane.

Points are coplanar if they all lie in the same plane. Any three points are always coplanar (you can always find a plane through three points), but four or more points may or may not be coplanar.

These terms matter because many theorems depend on whether points share a line or a plane. For example, the Segment Addition Postulate requires the three points to be collinear — if the points are not on the same line, the postulate does not apply.

❓

Concept Check

Points A, B, and C are non-collinear. How many distinct planes contain all three points? What if a fourth point D is not on that plane — how many planes can you form using three of the four points?

▸

Exactly one plane contains three non-collinear points. With four points where D is not on the plane of A, B, and C, you can form four distinct planes: ABC, ABD, ACD, and BCD — each defined by choosing three of the four points.

Looking Ahead

These foundations — points, lines, angles, and constructions — are the raw materials for everything that follows. In Unit 2, you will learn how to combine these ideas with logical reasoning to write formal proofs. The vertical angles theorem, the properties of bisectors, and the postulates from this unit will become the building blocks of those arguments. Every proof in geometry ultimately traces back to the simple ideas you studied today.