ABBREVIATIONS.

18. The following is a list of the symbols which will be used as abbreviations:

In addition to these, the following are recommended for writing demonstrations on the board or in exercise books, but no use is made of them in the present work:

PLANE GEOMETRY.

BOOK I.

RECTILINEAR FIGURES.

DEFINITIONS AND GENERAL PRINCIPLES.

19. When two straight lines meet at a point, they are said to form an angle with each other.

The point of meeting of the lines is called the vertex of the angle, and the lines are called its

sides.

Thus in the angle formed by AB and BC, B is the vertex and AB and BC are the sides.

B

A

20. When there is but one angle at a given vertex, it may be designated by the letter at that point; but if several angles have a common vertex, it is necessary in order to avoid ambiguity to name also the letters at the extremities of the sides, placing the letter at the vertex between the others.

Thus we should designate the angle of the preceding article as "the angle B"; but if there were other angles at the same vertex, we should read it either ABC or CBA.

21. The magnitude of an angle depends solely on the amount of divergence of its sides, and is entirely independent of their length.

22. Two angles are equal when their vertices may be made to coincide in position and their sides in direction.

23. Two angles are called adjacent when they have the same vertex and a common side between them; as AOB and вос.

B

24. Through a given point within or without a given straight line, a straight line may be drawn meeting the given line in such a way as to make the adjacent angles equal.

Each of the equal angles is called a right angle, and the lines are said to be perpendicular to each other.

Thus if AB is drawn so as to make the adjacent angles BAC and BAD equal, each of these angles is a right angle and the lines AB and CD are perpendicular to each other.

B

C

A

25. At a given point in a straight line but one perpendicular to that line can be drawn.

For if the line AB were to be revolved about the point A as a pivot, one of the equal angles would increase and the other would diminish; and there would evidently be but one position in which they would be equal.

26. An acute angle is one which is less than a right angle; as ABC.

An obtuse angle is one which is greater than a right angle; as DEF.

Acute and obtuse angles are called oblique angles, and intersecting lines which are not perpendicular are called oblique to each other.

27. The magnitude of an angle is measured by finding the number of times which it contains another angle adopted arbitrarily as the unit of measurement.

The usual unit of measurement is the degree, or the ninetieth part of a right angle. To express fractional parts of the unit, the degree is divided into sixty equal parts, called minutes, and the minute into sixty equal parts, called seconds.

Degrees, minutes, and seconds are denoted by the symbolso respectively; thus, 43° 22′ 37′′ denotes an angle

of 43 degrees, 22 minutes, and 37 seconds.

28. If the sum of two angles is a right angle, or 90°, one is called the complement of the other; if their sum is two right angles, or 180°, they are called supplements of each other.

Thus, 30° and 60° are complements of each other, and 30° and 150° are supplements.

29. It is evident that the complements of equal angles are equal to each other, and the same is true of their supplements.

30. Two angles are called vertical, or A opposite, when the sides of one are the prolongations of the sides of the other; as AOC and BOD.

B

EXERCISES.

1. How many degrees are there in the complement of 34°? of 88°? of 90°?

2. How many degrees are there in the supplement of 45°? of 90°? of 168°?

3. How many degrees are there in the complement, and in the supplement, of an angle equal to of a right angle?

4. How many degrees are there in the angle whose supplement is equal to three times its complement?

PERPENDICULARS AND OBLIQUE LINES.

PROPOSITION I. THEOREM.

31. If one straight line meet another, the sum of the two adjacent angles formed is equal to two right angles.

Let the straight line CD meet AB at C.

To prove that the sum of the angles ACD and BCD is equal to two right angles.

Draw CE perpendicular to AB at C.

Then it is evident that the sum of the angles ACD and BCD is equal to the sum of the angles ACE and BCE, each of which is a right angle.

Hence the sum of the angles ACD and BCD is equal to two right angles.

32. SCHOLIUM. By § 28 each of the angles ACD and BCD is the supplement of the other; such angles are called supplementary-adjacent.

33. COROLLARY I. The sum of all the angles AOB, BOC, COD, formed on the same side of a straight line at a given point, is equal to two right angles.

D

B

For this sum is equal to the sum of the two adjacent angles AOB and BOD, which is two right angles by § 31.