Addition of Lines. Having defined the equality of straight lines, we proceed to explain what is meant by the addition of straight lines.

If in a straight line we take points A, B, C, D in order, we say that the straight line AC is the sum of the two straight lines AB, BC (or of any two straight lines equal to them),

and that the straight line AB is the difference of the two straight lines AC, BC (or of any two straight lines equal to them).

In the same way we say that the straight line AD is the sum of the three straight lines AB, BC, CD.

Again, if AB be equal to BC, we say that AC is double of AB or of BC.

DEFINITION 7. A surface which lies evenly between straight lines on it is called a plane.

This is Euclid's definition of a plane: there is the same difficulty in making use of it that there is in making use of his definition of a straight line.

Consequently this definition has by many modern editors been replaced by the following, which perhaps merely expresses Euclid's meaning in other words:

A surface such that the straight line joining any two points in the surface lies wholly in the surface is called a plane.

DEFINITION 8. A figure, which lies wholly in one plane, is called a plane figure.

All the geometrical propositions in the first six books of the Elements of Euclid relate to figures in one plane. This part of Geometry is called Plane Geometry.

DEFINITION 9. Two straight lines in the same plane, which do not meet however far they may be produced both ways, are said to be parallel* to one another.

A

B

D

DEFINITION 10. A plane angle is the inclination to one another of two straight lines which meet but are not in the same straight line.

The idea of an angle is one which it is very difficult to convey by the words of a definition. We will content ourselves by explaining some few things connected with angles.

If two straight lines AB, AC meet at A, the amount of their divergence from one another or their inclination to one another is called the angle which the lines make with one another or the angle between the lines, or the angle contained by the lines.

A

B

A B

The angle formed by the straight lines AB, AC is generally denominated BAC, or CAB, the middle letter always denoting the point where the lines meet, and the letters B and C denoting any two points in the straight lines AB, AC. It must be carefully noted that the magnitude of the angle is not affected by the length of the straight lines AB, AC.

The point A, where the two straight lines AB and AC, which form the angle BAC, meet, is called the vertex of the angle BAC.

If there be only two straight lines meeting at a point A, the angle formed by the lines is sometimes denoted by the single letter A.

* Derived from Tapá "by the side of" and dλλýλas “one another": παράλληλοι γραμμαί “lines side by side”.

Test of Equality of Angles. Two angles are said to be equal, when it is possible to shift the straight lines forming one of the angles, unchanged in position relative to each other, so as to exactly coincide in direction with the straight lines forming the other angle. D)

For instance, the angles ABC, DEF will be equal, if it be possible to shift AB, BC unchanged in position relative to each other, so that B coincides with E, and so that also either BA coincides in direction with ED and BC with EF, or BA coincides in direction with EF and BC with ED.

If a straight line move in a plane, while one point in the line remains fixed, the line is said to turn or revolve about the fixed point. If the revolving line move from any one position to any other position, it generates an angle, and the amount of turning from one position to the other is the measure of the magnitude of the angle between the two positions of the line.

For instance each hand of a watch, as long as the watch is going, is turning uniformly round its fixed extremity, and is generating an angle uniformly.

This mode of regarding angles enables us to realize that angles are capable of growing to any size and need not be limited (as in most of the propositions in Euclid's Elements they are supposed to be) to magnitudes less than two right angles. (See Def. 11.)

Addition of Angles. If three straight

lines AB, AC, AD meet at the same point, we say that the angle BAD is the sum of the two angles BAC, CAD (or of any two angles equal to them).

In the same way we say that the angle BAC is the difference of the two angles

BAD, CAD (or of any two angles equal to them).

B

Two angles such as BAC, CAD, which have a common vertex and

one common bounding line, are called adjacent angles.

A

DEFINITION 11. If two adjacent angles made by two straight lines at the point where they meet be equal, each of these angles is called a right angle, and the straight lines are said to be at right angles to each other.

Either of two straight lines which are at right angles to each other is said to be perpendicular to the other.

E

D

B

If a straight line AE be drawn from a point A at right angles to a given straight line CD, the part AE intercepted between the point and the straight line is commonly called the perpendicular from the point A on the straight line CD.

Euclid uses as a postulate,

POSTULATE 5. All right angles are equal to one another. It is not necessary to assume this proposition, since it can be proved by the method of superposition. A proof will be found on a subsequent page. (p. 37)

DEFINITION 12. An angle less than a right angle is called an acute angle.

An angle greater than a right angle and less than two right angles is called an obtuse angle.

DEFINITION 13. A line, which is such that it can be described by a moving point starting from any point of the line and returning to it again, is called a closed line.

A figure composed wholly of straight lines is called a rectilineal figure.

The straight lines, which form a closed rectilineal figure, are called the sides of the figure.

The sum of the lengths of the sides of any figure is called the perimeter of the figure.

The point, where two adjacent sides meet, is called a vertex or an angular point of the figure.

The angle formed by two adjacent sides is called an angle of the figure.

Test of Equality of Angles. Two angles are said to be equal, when it is possible to shift the straight lines forming one of the angles, unchanged in position relative to each other, so as to exactly coincide in direction with the straight lines forming the other angle. D

A

E

B

For instance, the angles ABC, DEF will be equal, if it be possible to shift AB, BC unchanged in position relative to each other, so that B coincides with E, and so that also either BA coincides in direction with ED and BC with EF, or BA coincides in direction with EF and BC with ED.

If a straight line move in a plane, while one point in the line remains fixed, the line is said to turn or revolve about the fixed point. If the revolving line move from any one position to any other position, it generates an angle, and the amount of turning from one position to the other is the measure of the magnitude of the angle between the two positions of the line.

For instance each hand of a watch, as long as the watch is going, is turning uniformly round its fixed extremity, and is generating an angle uniformly.

This mode of regarding angles enables us to realize that angles are capable of growing to any size and need not be limited (as in most of the propositions in Euclid's Elements they are supposed to be) to magnitudes less than two right angles. (See Def. 11.)

Addition of Angles. If three straight

lines AB, AC, AD meet at the same point, we say that the angle BAD is the sum of the two angles BAC, CAD (or of any two angles equal to them).

In the same way we say that the angle BAC is the difference of the two angles

A

BAD, CAD (or of any two angles equal to them).

B

Two angles such as BAC, CAD, which have a common vertex and

one common bounding line, are called adjacent angles.