Some of Euclid's Book 1 Definitions 1. A point is that which has no part. This can be interpreted to mean that a point is 2. A line is breadthless length. A line is a construct that has no thickness. It can be considered as a continuous succession of points. 4. A straight line is a line which lies evenly with the points on itself. 8. A plane angle is the inclination to one another of two lines in a plane which meet one another and do not lie in a straight line. Thus, the amount of rotation about the intersection required to bring one line into correspondence with the other is the angl between the lines. 10. Right Angle / Perpendicular Lines: When a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is and the straight line standing on the other is called a In the diagram, the line CD has been constructed so as to be a perpendicular to the line AB 15. A circle is a plane figure contained by one line such that all the straight line falling upon it from one point among those lying within the figure are equal to one another. 16. And the point is called the center of the circle

Some of Euclid's Book 1 Definitions

This can be interpreted to mean that a point is indivisible: it cannot be partitioned into anything smaller.

that has no thickness. It can be considered as a continuous succession of points.

is a line which lies evenly with the points on itself.

is the inclination to one another of two lines in a plane which meet one

Thus, the amount of rotation about the intersection required to bring one line into correspondence with the other is the angl

When a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is and the straight line standing on the other is called a perpendicular to that on which it stands.

has been constructed AB.

is a plane figure contained by one line such that all the straight line [segments] upon it from one point among those lying within the figure are equal to one

center of the circle.

2.1

Thus, the amount of rotation about the intersection required to bring one line into correspondence with the other is the angle

When a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is right,

2.2

Euclid's Common Notions This is a set of axiomatic statements that appear at the start of Book I of The Elements by Euclid. 1. Things which are equal to the same thing are also equal to each other. 2. If equals are added to equals, the wholes are equal. 3. If equals are subtracted from equals, the remainders are equal. 4. Things which coincide with one another are equal to one another. 5. The whole is greater than the part.

Euclidean Propositions 1. Construction of Equilateral Triangle Theorem On a given straight line segment, it is possible to construct an equilateral triangle. Construction Let AB be the given straight line segment. We construct a circle BCD with center A and radius We construct a circle ACE with center B and radius From C, where the circles intersect, we draw a line segment to the straight line segments AC and BC. Then ABC is the equilateral triangle required. Proof As A is the center of circle BCD, it follows from As B is the center of circle ACE, it follows from So, as AC = AB and BC = AB, it follows from Therefore AB = AC = BC. Therefore ABC is equilateral. ■ Historical Note This is Proposition 1 of Book I of Euclid's The Elements

1. Construction of Equilateral Triangle

On a given straight line segment, it is possible to construct an equilateral triangle.

and radius AB. and radius AB.

, where the circles intersect, we draw a line segment to A and to B to form

is the equilateral triangle required.

, it follows from Book I Definition 15: Circle that AC = AB. , it follows from Book I Definition 15: Circle that BC = AB.

, it follows from Common Notion 1 that AC = BC.

The Elements.

2.3

2. Triangle Side-Angle-Side Equality Implies Congruence Theorem If two triangles have:

two sides equal to two sides respectively; the angles contained by the equal straight lines equal

they will also have: their third sides equal; the remaining two angles equal to their respective remaining angles, namely, those which the equal sides subtend

Proof Let ABC and DEF be two triangles having sides AB = DE and AC = DF , and with ∠ BAC = ∠ If ABC is placed on DEF such that: the point A is placed on point D, and the line the point B will also coincide with point E because So, with AB coinciding with DE , the line ACDF because ∠ BAC = ∠ EDF. Hence the point C will also coincide with the point But B also coincided with E. Hence the line BC will coincide with line EF. (Otherwise, when B coincides with E and Ccoincide with line EF and two straight lines will enclose a space which is impossible.) Therefore BC will coincide with EF and be equal to it. Thus the whole ABC will coincide with the whole The remaining angles on ABC will coincide with the remaining angles on ■ Historical Note This is Proposition 4 of Book I of Euclid's The Elements

Side Equality Implies Congruence

two sides equal to two sides respectively; the angles contained by the equal straight lines equal

remaining two angles equal to their respective remaining angles, namely, those which the equal sides subtend

be two triangles having sides ∠ EDF.

, and the line AB is placed on line DE then because AB = DE.

AC will coincide with the line

will also coincide with the point F , because AC = DF.

C with F, the line BC will not

and two straight lines will enclose a space which is impossible.) and be equal to it.

will coincide with the whole DEF and thus ABC = DEF. will coincide with the remaining angles on DEF and be equal to them.

The Elements.

2.4

remaining two angles equal to their respective remaining angles, namely, those which the equal sides subtend.

5. Bisection of an Angle Construction Let ∠ BAC be the given angle to be bisected. Take any point D on AB. We cut off from AC a length AE equal to ADWe draw the line segment DE. We construct an equilateral triangle DEF on We draw the line segment AF. Then the angle ∠ BAC has been bisected by the straight line segment Proof We have: AD = AE; AF is common; DF = EF. Thus triangles ADF and AEF are equal. Thus ∠ DAF = ∠ EAF. Hence ∠ BAC has been bisected by AF. ■ Historical Note This is Proposition 9 of Book I of Euclid's The Elements There are quicker and easier constructions of a bisection, but this particular one uses only results previously demonstrated.

be the given angle to be bisected.

AD.

on AB.

has been bisected by the straight line segment AF.

The Elements.

There are quicker and easier constructions of a bisection, but this particular one uses only results previously demonstrated.

2.7

There are quicker and easier constructions of a bisection, but this particular one uses only results previously demonstrated.

Internal/External Angle Definitions ■ Internal Angle The internal angle (or interior angle) of a vertex is the size of the angle between the sides forming that vertex, as measured inside the polygon. ■ External Angle Surprisingly, the external angle (or exterior anglebetween the sides forming that vertex, as measured outside the polygon. It is in fact an angle formed by one side of a polygon and a line produced from an adjacent side. While ∠ AFE is the internal angle of vertex F Note: it doesn't matter which adjacent side you use, since they are equal by the

) of a vertex is the size of the angle between the the polygon.

exterior angle) of a vertex is not the size of the angle between the sides forming that vertex, as measured outside the polygon. It is in fact an angle formed by one side of a polygon and a line produced from an adjacent

F , the external angle of this vertex is ∠ EFG. Note: it doesn't matter which adjacent side you use, since they are equal by the Vertical Angle Theorem.

2.8

9. External Angle of Triangle Greater than Theorem The external angle of a triangle is greater than either of the opposite internal angles. Proof Let ABC be a triangle. Let the side BC be extended to D. Let AC be bisected at E. Let BE be joined and extended to F. Let EF be made equal to BE. (Technically we really need to extend BE to a point beyond length EF.) Let CF be joined. Let AC be extended to G. have ∠ AEB = ∠ CEF from Two Straight Lines make Equal Opposite Angles. Since AE = EC and BE = EF , from Triangle SideABE = CFE. Thus AB = CF and ∠ BAE = ∠ ECF. But ∠ ECD is greater than ∠ ECF. Therefore ∠ ACD is greater than ∠ BAE. Similarly, if BC were bisected, ∠ BCG, which is equal to be greater than ∠ ABC as well. Hence the result. ■ Historical Note This is Proposition 16 of Book I of Euclid's The Elements

External Angle of Triangle Greater than Internal Opposite Angles

is greater than either of the opposite internal angles.

to a point beyond F and then crimp off

Two Straight Lines make Equal Opposite Angles. Triangle Side-Angle-Side Equality we have

, which is equal to ∠ ACD by Two Straight Lines make Equal Opposite Angles

The Elements.

2.9

a

We

Two Straight Lines make Equal Opposite Angles, would be shown to

16. Equal Alternate Interior Angles Implies Parallel Theorem Given two infinite straight lines which are cut by a transversal, if the alternate interior angles are equal, then the lines Proof Let AB and CD be two straight lines, and let transversal that cuts them. Let at least one pair of alternate interior angles, be equal. Without loss of generality, let ∠ AEF = ∠ EFD. Assume that the lines are not parallel. Then they meet at some point G which, without loss of generality, is on the same side as B and D. Since ∠ AEF is an exterior angle of GEF, then from External Angle of Triangle Greater than Internal Opposite∠ AEF > ∠ EFG, a contradiction. Similarly, they cannot meet on the side of A and Therefore, by definition, they are parallel. ■ Historical Note This is Proposition 27 of Book I of Euclid's The This theorem is the converse of the first part of Proposition 29.

Equal Alternate Interior Angles Implies Parallel

Given two infinite straight lines which are cut by a transversal, if the alternate interior angles are equal, then the lines

be two straight lines, and let EF be a transversal that cuts them. Let at least one pair of alternate interior angles, be equal. Without loss of generality, let

Assume that the lines are not parallel. which, without loss of

, then from External Angle of Triangle Greater than Internal Opposite,

and C.

The Elements. This theorem is the converse of the first part of Proposition 29.

2.12

Given two infinite straight lines which are cut by a transversal, if the alternate interior angles are equal, then the lines are parallel.

17. Equal Corresponding Angles or Supplementary Interior Angles Implies Parallel Theorem Part 1 Given two infinite straight lines which are cut by a transversal, if the corresponding angles are equal, then the lines are parallel. Part 2 Given two infinite straight lines which are cut by a transversal, if the interior angles on the same side of the transversal are supplementary, then the lines are parallel. Proof Part 1 Let AB and CD be infinite straight lines, and let them. Let at least one pair of corresponding angles be equal. Without loss, let ∠ EGB and ∠ GHD be equal. Then ∠ GHD = ∠ EGB = ∠ AGH by the Vertical Angle Theorem.Thus AB ∥ CD by Equal Alternate Interior Angles ■ Part 2 Let AB and CD be infinite straight lines, and let the transversal be supplementary. Without loss, let right angles. ∠ AGH + ∠ BGH equals two right angles. Then from Euclid's first and third common notions ∠ AGH = ∠ DHG. Finally, AB ∥ CD by Equal Alternate Interior Angles Implies Parallel■ Historical Note This is Proposition 28 of Book I of Euclid's The ElementsThis theorem is the converse of the second and third parts of Proposition 29.

Equal Corresponding Angles or Supplementary Interior Angles Implies Parallel

Given two infinite straight lines which are cut by a transversal, if the corresponding

Given two infinite straight lines which are cut by a transversal, if the interior angles on the same side of the transversal are supplementary, then the lines are parallel.

be infinite straight lines, and let EF be a transversal that cuts them. Let at least one pair of corresponding angles be equal. Without loss, let

Vertical Angle Theorem. Equal Alternate Interior Angles Implies Parallel.

be infinite straight lines, and let EF be a transversal that cuts them. Let at least one pair of interior angles on the same side of the transversal be supplementary. Without loss, let ∠ BGH and ∠ DHG be supplementary, so by definition

first and third common notions and Euclid's fourth postulate,

Interior Angles Implies Parallel.

The Elements. This theorem is the converse of the second and third parts of Proposition 29.

2.13

be a transversal that cuts them. Let at least one pair of interior angles on the same side of lementary, so by definition ∠ DHG + ∠ BGH equals two

18. Parallel Implies Equal Alternate Interior Angles, Corresponding Angles, and Supplementary Interior Angles

Part 1 Given two infinite straight lines which are cut by a transversal, if the lines are parallel, then the alternate interior angles are equal. Part 2 Given two infinite straight lines which are cut by a transversal, if the lines are parallel, then the corresponding angles are equal. Part 3 Given two infinite straight lines which are cut by a then the interior angles on the same side of the transversal a Proof Let AB and CD be parallel infinite straight lines, and let cuts them.

Part 1 Assume the alternate interior angles are not equal. Then one of the pair ∠ AGH be greater. Now ∠ AGH + ∠ BGH equal two right angles, so ∠ GHD + ∠ BGH is less than two right angles. Lines extended infinitely from angles less than two right angles must meet due to Euclid's fifth postulate. But the lines ardefinition the lines do not intersect. This is a contradiction. Thus, the alternate interior angles must be equal. ■

Parallel Implies Equal Alternate Interior Angles, Corresponding Angles, and Supplementary Interior Angles

Given two infinite straight lines which are cut by a transversal, if the lines are parallel,

Given two infinite straight lines which are cut by a transversal, if the lines are parallel,

which are cut by a transversal, if the lines are parallel, on the same side of the transversal are supplementary.

be parallel infinite straight lines, and let EF be a transversal that

Assume the alternate interior angles are not equal. Then one of the pair ∠ AGH and ∠ GHD must be greater. Without loss of generality, let

equal two right angles, ight angles.

Lines extended infinitely from angles less than two right angles must meet due to Euclid's fifth postulate. But the lines ardefinition the lines do not intersect. This is a contradiction.

es must be equal.

2.14

must be greater. Without loss of generality, let

Lines extended infinitely from angles less than two right angles must meet due to Euclid's fifth postulate. But the lines are parallel, so by

Part 2 From part 1, ∠ AGH = ∠ DHG. So ∠ EGB = ∠ AGH = ∠ DHG due to the two straight lines cut each other, they make the opposite angles equal each other. ■ Part 3 From part 2 and Euclid's second common notion, ∠ EGB + ∠ BGH = ∠ DHG + ∠ BGHFurther, ∠ EGB + ∠ BGH equal two right angles, so by definition ∠ BGH and ∠ DHG are supplementaryI.e., when ∠ BGH and ∠ DHG are set next to each other, they form a straight angle. ■ Note: This is Proposition 29 of Book I of Euclid's is the first proposition to make use of Euclid's fifth postulate.

due to the Vertical Angle Theorem: If two straight lines cut each other, they make the opposite angles equal each

From part 2 and Euclid's second common notion, BGH.

equal two right angles, supplementary.

to each other, they form a straight

Proposition 29 of Book I of Euclid's The Elements. Proposition 29 is the first proposition to make use of Euclid's fifth postulate.

2.15

Area in Geometry Area is a quantity expressing the size of a figure in the surface. Points and lines have zero area, although there are the particular definition taken, a figure may have infinite area, for example the entire Euclidean plane. In three dimensions, the analog of area is called a How to define area

Although area seems to be one of the basic notions in geometry, it is noeven in the Euclidean plane. Most textbooks avoid defining an area, relying on selfTo make the concept of area meaningful one has to define it, at the very least, on polygons in the Euclidean plane, and it can be done using the following definition:

The area of a polygon in the Euclidean plane is a positive number such that:

1. The area of the unit square is equal to one. 2. Congruent polygons have equal areas. 3. Additivity: If a polygon is a union of two polygons which do not have common interior points, then its area is the sum of the areas of these two polygons. The area of an arbitrary square

(i) If 𝑛 is any positive integer ( i.e. a “whole number” ) from the additivity property of area that the A = 𝑛 𝑠𝑞𝑢𝑎𝑟𝑒 𝑢𝑛𝑖𝑡𝑠. .

Proof

Observe that there are:

2 . . . nnnn terms

nnn

unit square tiles which cover the square with side length 𝑛 units.

A = 𝑛 × (1 𝑢𝑛𝑖𝑡 ) = 𝑛 𝑠𝑞𝑢𝑎𝑟𝑒

expressing the size of a figure in the Euclidean plane or on a 2-dimensional . Points and lines have zero area, although there are space-filling curves. Depending on

particular definition taken, a figure may have infinite area, for example the entire Euclidean plane. In three dimensions, the analog of area is called a volume.

Although area seems to be one of the basic notions in geometry, it is not at all easy to define even in the Euclidean plane. Most textbooks avoid defining an area, relying on self-evidence. To make the concept of area meaningful one has to define it, at the very least, on polygons in

using the following definition:

The area of a polygon in the Euclidean plane is a positive number such that:

The area of the unit square is equal to one.

If a polygon is a union of two polygons which do not have common interior points, then its area is the sum of the areas

is any positive integer ( i.e. a “whole number” ) then it follows immediately additivity property of area that the square with side length 𝑛 has area:

𝑠𝑞𝑢𝑎𝑟𝑒 𝑢𝑛𝑖𝑡𝑠.

2.20

If a polygon is a union of two polygons which do not have common interior points, then its area is the sum of the areas

2.21

(ii) If 𝑛 is any positive integer then it also follows from the additivity property

that the square with side length units must have area:

A =

𝑠𝑞𝑢𝑎𝑟𝑒 𝑢𝑛𝑖𝑡𝑠.

Proof

Observe that there are 𝑛 square tiles with side lengths units which

cover the unit square.

𝑛 × A = 1 𝑢𝑛𝑖𝑡

A =

𝑠𝑞𝑢𝑎𝑟𝑒 𝑢𝑛𝑖𝑡𝑠.

(iii) For any positive integers 𝑚 and 𝑛 , the square with side lengths units

has area A = 𝑠𝑞𝑢𝑎𝑟𝑒 𝑢𝑛𝑖𝑡𝑠.

We may obtain this result by additivity once again.

Proof

In this case, note that there are 𝑛 square tiles of side length units

which cover the square having side length 𝑚 units.

𝑛 × A = 𝑚 𝑢𝑛𝑖𝑡𝑠

A = 𝑠𝑞𝑢𝑎𝑟𝑒 𝑢𝑛𝑖𝑡𝑠.

2.22

(iv) For any positive real number 𝑠 the area of the square with side length 𝑠 is A = 𝑠 𝑠𝑞𝑢𝑎𝑟𝑒 𝑢𝑛𝑖𝑡𝑠. This follows from the well known density property that any positive number 𝑠 (whether rational or irrational)

may be expressed as the limit of a sequence of numbers of the form where 𝑚 and 𝑛 are positive integers

for all 𝑖 = 1 , 2 , 3, . . . Thus, the difference: 𝑠 − 𝑚𝑖

𝑛𝑖 becomes infinitesimally small as 𝑖 becomes larger and larger.

That is, there exists a sequence:

𝑚

𝑛,

𝑚

𝑛,

𝑚

𝑛, . . . ,

𝑚

𝑛 → 𝑠 as 𝑖 increases without bound.

Proof

In the figure, we have → 𝑠 .

So due to the continuity of geometrical area, it must be the case that

𝐀𝒊 =𝑚𝑖

𝑛𝑖×

𝑚𝑖

𝑛𝑖 𝑢𝑛𝑖𝑡𝑠 =

𝑚𝑖

𝑛𝑖𝑢𝑛𝑖𝑡𝑠 → 𝐀

as 𝑖 increases without bound.

A = 𝑠 𝑠𝑞𝑢𝑎𝑟𝑒 𝑢𝑛𝑖𝑡𝑠..