Euclidean Geometry

Deductive geometry founded on axioms

Goal of this page

Understand the axiom system of Euclidean geometry and learn the concepts of congruence and similarity rigorously. Acquire the deductive method of deriving theorems from axioms.

1. Euclid's axioms

Euclidean geometry is the classical geometry that treats figures in the plane and in space using only logic built on a small set of axioms (postulates). Most of the properties of figures taught in secondary school (middle and high school) belong to this framework. In contrast to analytic geometry, which foregrounds coordinates and computation, it derives the properties of figures one by one through construction and logical argument.

At the beginning of Book I of the Elements, Euclid built plane geometry on the basis of five postulates (aitēmata) and five common notions. The five items listed below are the postulates (Book I, Postulates 1–5).

Euclid's five postulates (Elements, Book I, Postulates 1–5)
  1. A straight line can be drawn joining any two points.
  2. A segment can be extended to any length.
  3. A circle can be described with any point as center and any radius.
  4. All right angles are equal to one another.
  5. Parallel postulate: through a point not on a given line, there is exactly one line parallel to it.
ℓ P m ℓ ‖ m
Figure 1. The parallel postulate. Through a point $P$ not on the line $\ell$, there is exactly one line $m$ parallel to $\ell$.

The fifth postulate is independent of the others, and negating it yields non-Euclidean geometry. In hyperbolic geometry, for instance, there are infinitely many lines through a point parallel to a given line, while in elliptic geometry there are none.

A note on terminology and wording.

  • Postulates vs. axioms (common notions): the Elements clearly distinguishes the postulates proper to geometry from the common notions concerning quantity in general (for example, "things equal to the same thing are equal to one another"). In everyday usage both are often lumped together under the word "axiom", but they are originally distinct.
  • The form of the fifth postulate: the statement given above, "there is exactly one parallel through a point", is not Euclid's own fifth postulate but the equivalent Playfair's axiom (J. Playfair, 1795). Euclid's original wording asserts the existence of an intersection: "if a straight line crossing two lines makes the interior angles on one side sum to less than two right angles, then the two lines meet on that side."

2. Congruence of triangles

Definition: congruence

Two figures are congruent if one can be translated, rotated, or reflected (a rigid motion) so as to coincide exactly with the other.

Notation: $\triangle ABC \cong \triangle DEF$

* The word "congruent" and the phrase "move to coincide" are modern textbook style. The Elements has no such word; in Proposition 4 it shows that sides and angles are "equal" by actually superposing the figures (the principle of superposition).

Theorem: congruence conditions for triangles

If any one of the following holds, the two triangles are congruent:

  1. SSS: the three sides are respectively equal (Elements, Book I, Prop. 8).
  2. SAS: two sides and the included angle are respectively equal (Book I, Prop. 4).
  3. ASA: one side and the two adjacent angles are respectively equal (Book I, Prop. 26, which also covers AAS).

* In school these three are usually taught together with two more, as five congruence conditions: AAS (two angles and a non-included side, contained in Book I, Prop. 26) and the right-triangle condition hypotenuse and one leg (HL; the remaining leg is then fixed by the Pythagorean theorem). SSS, SAS, and ASA are the basic forms (see also the FAQ at the end).

a b c SSS SAS ASA
Figure 2. Congruence conditions for triangles. If any one of SSS (three sides), SAS (two sides and the included angle), or ASA (one side and the two adjacent angles) holds, congruence is determined. The thick colored edges mark the "sides of interest" assumed equal in each condition (the arcs mark the angles of interest).

Idea of the proof (SAS = Prop. 4) and "superposition"

The proof of Proposition 4 (SAS) in Book I of the Elements begins by moving one triangle $\triangle ABC$ so as to superpose it on the other $\triangle DEF$. Placing $A$ on $D$ and side $AB$ along $DE$, since $AB=DE$ and $\angle A=\angle D$ the point $B$ falls on $E$ and side $AC$ lies along $DF$. Because $AC=DF$, the point $C$ falls on $F$, so $BC$ coincides with $EF$ and the two triangles coincide completely. $\square$

* This "superposition" tacitly assumes that a figure keeps its shape and size when moved, which is not stated as a postulate. The gap was later removed by Hilbert (1899, Foundations of Geometry), who recast congruence as an independent axiom. SSS (Prop. 8) and ASA (Prop. 26) are proved on the basis of this Proposition 4.

3. Similarity of triangles

Definition: similarity

Two figures are similar if one can be enlarged or shrunk so as to become congruent to the other.

Notation: $\triangle ABC \sim \triangle DEF$

* The phrase "enlarge or shrink to make congruent" is modern textbook style. In the Elements, Book VI, Definition 1, similar figures are defined as those "whose corresponding angles are respectively equal and whose corresponding sides are proportional"; the language of scaling (a similarity transformation) is not used.

Theorem: similarity conditions for triangles
  1. SSS similarity: all three pairs of sides are in the same ratio (Elements, Book VI, Prop. 5).
  2. SAS similarity: two pairs of sides are in the same ratio and the included angles are equal (Book VI, Prop. 6).
  3. AA: two angles are respectively equal (Book VI, Prop. 4).

* In order, these correspond to the congruence conditions SSS, SAS, ASA (three side-ratios / two side-ratios with the included angle / two angles). Where congruence asks for "equal sides," similarity asks for "sides in equal ratio."

∼ SSS ∼ SAS ∼ AA
Figure 3. Similarity conditions for triangles. If any one of SSS similarity (three sides in proportion), SAS similarity (two sides in proportion and the included angle equal), or AA (two equal angles) holds, the two triangles are similar. The thick colored edges mark the sides assumed to be in proportion, and the arcs mark the angles assumed equal; each panel shows a small triangle and a similar larger one.
Theorem: similarity ratio and area ratio (Elements, Book VI, Prop. 19)

When the similarity ratio is $k : 1$:

  • the ratio of corresponding sides is $k : 1$;
  • the ratio of areas is $k^2 : 1$.

* This corresponds to Book VI, Proposition 19: "similar triangles are to one another in the duplicate ratio of their corresponding sides."

S k = 1 ×2 S S S S k = 2
Figure 4. Similarity ratio and area ratio. Scaling by a similarity ratio of $k=2$ splits the larger triangle into $4$ small triangles congruent to the original, so the ratio of areas is $k^2 = 4$.

Proof sketch (area ratio = Prop. 19)

In similar triangles with similarity ratio $k:1$, corresponding angles are equal, so a corresponding base is $k$ times as long and the height to that base is also $k$ times as long. Since the area of a triangle is $\tfrac12\times{}$base${}\times{}$height, the area is multiplied by $k\times k=k^2$. $\square$

* As in Figure 4, the fact that a triangle scaled by $2$ splits into $4$ congruent pieces is a visual check of $k^2=4$ at $k=2$. Book VI, Proposition 19 of the Elements proves this rigorously using only the theory of ratios of areas, without coordinates.

4. Properties of circles

Theorem: the inscribed angle theorem (Elements, Book III, Prop. 20)

The inscribed angle subtending a given arc is constant and equals half the central angle.

$$\angle APB = \dfrac{1}{2} \angle AOB$$
O A B P 2θ θ
Figure 5. The inscribed angle theorem. The inscribed angle $\angle APB=\theta$ subtending the arc $AB$ is exactly half the central angle $\angle AOB=2\theta$.

Proof (inscribed angle theorem = Prop. 20)

Extend the line joining the center $O$ and the point $P$ on the circle until it meets the circle again at $Q$ (so $PQ$ is a diameter).

Since $OA=OP$ (both radii), $\triangle OAP$ is isosceles and its base angles are equal, $\angle OPA=\angle OAP$. An exterior angle of a triangle equals the sum of the two non-adjacent interior angles, so $$\angle AOQ=\angle OPA+\angle OAP=2\,\angle OPA.$$ Likewise $OB=OP$ makes $\triangle OBP$ isosceles, giving $\angle BOQ=2\,\angle OPB$.

Adding the two, $$\angle AOB=\angle AOQ+\angle BOQ=2(\angle OPA+\angle OPB)=2\,\angle APB.$$ Hence $\angle APB=\tfrac12\,\angle AOB$. $\square$

* This shows the case where the center $O$ lies inside $\angle APB$. When $O$ lies outside the angle or on one of its sides, the same conclusion holds — one simply takes a difference instead of a sum, or one term becomes $0$.

O P Q A B
Figure 6. Proof of the inscribed angle theorem. Taking the diameter $PQ$ makes both $\triangle OAP$ and $\triangle OBP$ isosceles ($OA=OB=OP=$ radius), and the exterior-angle property yields $\angle AOB=2\,\angle APB$.
Theorem: the tangent-chord angle (Elements, Book III, Prop. 32)

The angle between a tangent to a circle and a chord equals the inscribed angle subtending that chord.

O tangent T chord B P α α
Figure 7. The tangent-chord angle. The angle $\alpha$ between the tangent at the point of tangency $T$ and the chord $TB$ equals the inscribed angle $\angle TPB=\alpha$ in the alternate segment.

Proof (tangent-chord angle = Prop. 32, from the inscribed angle theorem)

Let $O$ be the center and $r$ the radius. Since $OT=OB=r$, $\triangle OTB$ is isosceles; writing the central angle as $\angle TOB=\beta$, its base angle is $\angle OTB=90^\circ-\tfrac{\beta}{2}$. The tangent is perpendicular to the radius at the point of tangency ($\angle OTA=90^\circ$), so the tangent-chord angle is $$\alpha=90^\circ-\angle OTB=90^\circ-\left(90^\circ-\tfrac{\beta}{2}\right)=\tfrac{\beta}{2}.$$ On the other hand, the inscribed angle in the alternate segment subtending the chord $TB$ is, by the inscribed angle theorem, $\angle TPB=\tfrac12\angle TOB=\tfrac{\beta}{2}$. Hence $\alpha=\angle TPB$. $\square$

As this proof of the tangent-chord angle shows, many properties of circles are derived from the inscribed angle theorem as a starting point: the relation between an inscribed angle and the central angle is the key to the theorems about circles.

Summary

Key points of this chapter

  • Five postulates: the foundation of Euclidean geometry (especially the parallel postulate).
  • Congruence conditions: SSS, SAS, ASA.
  • Similarity conditions: AA, SAS similarity, SSS similarity.
  • Inscribed angle theorem: the inscribed angle subtending a given arc is constant.

Frequently asked questions

What are the axioms (postulates) of Euclidean geometry?

In the Elements, Euclid organized geometry on the basis of five postulates (1. a straight line can be drawn between any two points, 2. a segment can be extended, 3. any circle can be described, 4. all right angles are equal, 5. the parallel postulate). The independence of the fifth postulate gave rise to non-Euclidean geometries (hyperbolic and elliptic).

What are the five congruence conditions for triangles?

They are SAS (two sides and the included angle), ASA (two angles and the included side), AAS (two angles and a non-included side), SSS (three sides), and HL (the hypotenuse and one leg of a right triangle). When one of these holds, the two triangles are congruent (same size and shape). SSA is not in general a congruence condition (only the right-angle case is an exception) and is known as the "ambiguous case".

What is the difference between Euclidean geometry and analytic geometry?

Euclidean geometry uses a synthetic approach based on construction, axioms, and logical proof. Analytic geometry (Descartes) introduces a coordinate system, represents figures by equations, and handles them algebraically. The two are equivalent, but analytic geometry generalizes to higher dimensions more easily and is indispensable for computation in modern mathematics and physics.

What is Playfair's axiom?

It is the statement that "through a point not on a given line there is exactly one line parallel to it," a proposition equivalent to Euclid's fifth postulate (the parallel postulate). It is named after the 18th-century mathematician John Playfair. In modern textbooks the parallel postulate is usually presented in this form.

Why is SSA not a congruence condition?

Even if two sides and a non-included angle (SSA) are equal, the triangle is not always determined uniquely. The side opposite the given angle can meet the third side in two different ways, so two triangles of different shape can satisfy the same data (the ambiguous case). When the angle is right or obtuse the triangle is determined, and the right-angle case is exactly the right-triangle condition (hypotenuse and one leg, HL).