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31°. When two lines have not the same direction they are said to make an angle with one another, and an angle is a *c difference in direction.



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Illustration.-Let A and B

represent two stars, and E the A position of an observer's eye.


Since the lines EA and EB, which join the eye and the stars, have not the same direction they make an angle with one another at E.

1. If the stars appear to recede from one another, the angle at E becomes greater. Thus, if B moves into the position of C, the angle between EA and EC is greater than the angle between EA and EB.

Similarly, if the stars appear to approach one another, the angle at E becomes smaller; and if the stars become coincident, or situated in the same line through E, the angle at E vanishes.

Hence an angle is capable of continuous increase or diminution, and is therefore a magnitude. And, bẹing magnitudes, angles are capable of being compared with one another as to greatness, and hence, of being measured.

2. If B is moved to B', any point on EB, and A to A', any point on EA, the angle at E is not changed. Hence increasing or diminishing one or both of the segments which form an angle does not affect the magnitude of the angle.

Hence, also, there is no community in kind between an angle and a line-segment or a line.

Hence, also, an angle cannot be measured by means of line-segments or lines.

32°. Def.-A line which changes its direction in a plane while passing through a fixed point in the plane is said to rotate about the point.

The point about which the rotation takes place is the pole, and any segment of the rotating line, having the pole as an end-point, is a radius vector.

Let an inextensible thread fixed at O be kept stretched by a pencil at P. Then, when P moves, keeping the thread straight, OP becomes a radius vector rotating about the pole O.

When the vector rotates from direction OP to direction OP' it describes the angle between OP and OP'. Hence we have the following:--

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Def. 1.—The angle between two lines is the rotation necessary to bring one of the lines into the direction of the other.

The word "rotation," as employed in this definition, means the amount of turning effected, and not the process of turning.

Def. 2.—For convenience the lines OP and OP', which, by their difference in direction form the angle, are called the arms of the angle, and the point Ọ where the arms meet is the vertex.

Cor. From 31°, 2, an angle does not in any way depend upon the lengths of its arms, but only upon their relative directions.


33°. Notation of Angles.-1. The symbol is used for the word “angle.”


2. When two segments meet at a vertex the angle between them may be denoted by a single letter placed at the vertex, as the 40, or by a letter with or without an arch of dots, as ẞ; or by three letters of which the extreme ones denote points upon the arms of the angle and the middle one denotes the vertex, as LAOB.


3. The angle between two lines, when the vertex is not pictured, or not referred to, is expressed by (L. M), or ĹM, where L and M denote the lines in the one-letter notation (22°); or (AB, CD), where AB and CD denote the lines in the two-letter notation.

34°. Def.-Two angles are equal when the arms of the one may be made to coincide in direction respec· tively with the arms of the other; or when the angles are described by the same rotation.

Thus, if, when O' is placed upon O, and O'A' is made to lie along OA, O'B' can also be made to lie along OB, the ▲A'O'B' is equal to LAOB. This equality is symbolized thus: LA'O'B' = LAOB.



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Where the sign is to be interpreted as indicating the possibility of coincidence by superposition.


35°. Sum and Difference of Angles.-The sum of two angles is the angle described by a radius vector which describes the two angles, or their equals, in succession.


Thus if a radius vector starts from coincidence with OA and rotates into direction OP it describes the LAOP. If it next rotates into direction OP' it A describes the POP'. But in its whole

op P

rotation it has described the AOP'. Therefore,


Def. When two angles, as AOP and POP', have one arm in common lying between the remaining arms, the angles are adjacent angles.

36°. Def.—A radius vector which starts from any given direction and makes a complete rotation so as to return to its original direction describes a circumangle, or perigon.

One-half of a circumangle is a straight angle, and onefourth of a circumangle is a right angle.

37°. Theorem.-If any number of lines meet in a point, the sum of all the adjacent angles formed is a circumangle. OA, OB, OC, ..., OF are lines meeting in O. Then


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= a circumangle. D


Proof-A radius vector which starts from coincidence with OA and rotates into the successive directions, OB, OC, OF, OA describes in succession the angles AOB, BOC, EOF, FOA.

But in its complete rotation it describes a circumangle (36°). LAOB+BOC+...+4FOA= a circumangle.


Cor. The result may be thus stated :—

The sum of all the adjacent angles about a point in the plane is a circumangle.

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38°. Theorem.-The sum of all the adjacent angles on one side of a line, and about a point in the

line is a straight angle.

O is a point in the line AB; then
LAOC+4COB a straight angle.






Proof.—Let A and B be any two points

in the line, and let the figure formed by

AB and OC be revolved about AB without displacing the points A and B, so that OC may come into a position OC'. Then (24°, Cor. 2) O is not displaced by the revolution, LAOC=LAOC', and 4BOC=¿BOC';



and since the sum of the four angles is a circumangle (37°), therefore the sum of each pair is a straight angle (36°). q.e.d.

Cor. 1. The angle between the opposite directions of a line is a straight angle.

Cor. 2. If a radius vector be rotated until its direction is reversed it describes a straight angle. And conversely, if a radius vector describes a straight angle its original direction is reversed.

Thus, if OA rotates through a straight angle it comes into the direction OB. And conversely, if it rotates from direction OA to direction OB it describes a straight angle.



A Β ́ L





These four angles consist of two pairs of opposite or vertical angles, viz., A, A', and B, B', A being opposite A', and B being opposite B'.

39°. When two lines L and M cut one another four angles are formed about the point of intersection, any one of which may be taken to be the angle between the lines.

40°. Theorem.--The opposite angles of a pair formed by two intersecting lines are equal to one another.

¿A+¿B=a straight angle LA'+B=a straight angle.


(38°) (38°)


Def. 1.--Two angles which together make up a straight angle are supplementary to one another, and one is called the supplement of the other. Thus, A is the supplement of B', and B of A'.

Cor. If LA=4B, then LA'=4B=LB', and all four angles are equal, and each is a right angle (36°).

Therefore, if two adjacent angles formed by two intersecting lines are equal to each other, all four of the angles so formed are equal to one another, and each is a right angle.

Def. 2.--When two intersecting lines form a right angle at their point of intersection, they are said to be perpendicular to one another, and each is perpendicular to the other.

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