Equality and Equivalency are expressed by
Greater than, is expressed by

Thus: B is greater than A, is written
B is less than A,

A circle is expressed by

66

66

A triangle is expressed by

The term Hypothesis is expressed by

R.L

(Hy.) (Ax.)

The difference of two quantities, when it is not known which is the greater, is expressed by the symbol

Thus, the difference between A and B is written AN B..

BOOK I.

OF STRAIGHT LINES, ANGLES, AND POLYGONS.

THEOREM I.

When one straight line meets another, not at its extremity, the two angles thus formed are two right angles, or they are together equal to two right angles.

Let AB meet CD, and if AB is perpendicular to CD, it does not incline to either extremity of CD. In that case, the angle ABD is equal to the angle ABC, and is a right angle, by Definition 15.

B

D

But if these angles are unequal, we are to show that their sum is equal to two right angles. Conceive the line BE to be drawn from the point B, so as not to incline toward either extremity of CD; then, by Def. 15, the angles CBE and EBD are right angles; but the angles CBA and ABD make the same sum, or fill the same angular space, as the two angles CBE and EBD, and are, consequently, equal to two right angles. Hence the theorem; when one straight line meets another, not at its extremity, the sum of the two angles is equal to two right angles.

Cor. Hence, the two angles ABC and ABD are supplementary to each other, (Def. 21).

THEOREM II.

From any point in a straight line, not at its extremity, the sum of all the angles that can be formed on the same side of the line is equal to two right angles.

Let CD be any line, and B any point in it.

H

We are to show that the sum of all the angles which can be formed at B, on one side of CD, will be equal to two right angles.

E

By Th. 1, any two supplementary angles, as ABD, ABC, are together equal to two right angles. And since the angular space about the point B is neither increased nor diminished by the number of lines drawn from that point, the sum of all the angles DBA, ABE, EBH, HBC, fills the same spaces as any two angles HBD, HBC. Hence the theorem; from any point in a line, the sum of all the angles that can be formed on the same side of the line is equal to two right angles.

Cor. 1. And, as the sum of all the angles that can be formed on the other side of the line, CD, is also equal to two right angles; therefore, all the angles that can be formed quite round a point, B, by any number of lines, are together equal to four right angles.

Cor. 2. Hence, also, the whole circumference of a circle, being the sum of the measures of all the angles that can be made about the center F, (Def. 53), is the measure of four right angles; consequently, a semicircumference, is the mea

F

sure of two right angles; and a quadrant, or 90°, is the measure of one right angle.

THEOREM III.

If one straight line meets two other straight lines at a common point, forming two angles, which together are equal to two right angles, the two straight lines are one and the same line.

Let the line AB meet the lines BD and BE at the common point B, making the sum of the two angles ABD, ABE, equal to two right angles; we are to prove that DB and BE are one straight line.

E

If DB and BE are not in the same line, produce DB to C, thus forming one line, DBC.

Now by Th. 1, ABD + ABC must be equal to two right angles. But by hypothesis, ABD + ABE is equal to two right angles.

Therefore, ABD + ABC is equal to ABD + ABE, (Ax. 1). From each of these equals take away the common angle ABD, and the angle ABC will be equal to ABE, (Ax. 3). That is, the line BE must coincide with BC, and they will be in fact one and the same line, and they cannot be separated as is represented in the figure. Hence the theorem; if one line meets two other lines at a common point, forming two angles which together are equal to two right angles, the two lines are one and the same line.

THEOREM IV.

If two straight lines intersect each other, the opposite or vertical angles must be equal.

If AB and CD intersect each other at E, we are to demonstrate that the angle AEC is equal to the vertical angle DEB; and the angle AED, to the vertical angle CEB.

E

As AB is one line met by DE, another line, the two angles AED and DEB, on the same side of AB, are equal to two right angles, (Th. 1). Also, because CD is a right line, and AE meets it, the two angles AEC and AED are together equal to two right angles.

Therefore, AED + DEB = AEC + AED. (Ax. 1.) If from these equals we take away the common angle AED, the remaining angle DEB must be equal to the remaining angle AEC, (Ax. 3). In like manner, we ca prove that AED is equal to CEB. Hence the theore if the two lines intersect each other, the vertical angles m be equal.

Second Demonstration.

By Def. 11, the angle DEB is the difference in the direction of the lines ED and EB; and the angle AEC is the difference in the direction of the lines EC and EA. But ED is opposite in direction to EC; and EB is opposite in direction to EA.

Hence, the difference in the direction of ED and EB is the same as that of EC and EA, as is obvious by inspection.

Therefore, the angle DEB is equal to its opposite AEC. In like manner, we may prove AED = CEB. Hence the theorem; if two lines intersect each other, the vertical angles must be equal.

THEOREM V.

If a straight line intersects two parallel lines, the sum of the two interior angles on the same side of the intersecting line is equal to two right angles.

[NOTE. By interior angles, we mean angles which lie between the parallels; the exterior angles are those not between the parallels.]

Let the line EF intersect the parallels AB and CD; then we are to demonstrate that the angles BGH + GHD = 2 R. L

Because GB and HD are parallel, they are equally inclined to the line EF, or have

A

H

E

G

B

the same difference of direction from that line.

fore, the

There

FGB = GHD. To each of these equals add BGH, and we have FGB+BGH=GHD+BGH. But by Th. 1, the first member of this equation is equal to two right angles; and the second member is the sum of the two angles between the parallels. Hence the theorem; if a line intersects two parallel lines, the sum of the two interior angles on the same side of the intersecting line must be equal to two right angles.