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may also be multiplied and divided by a number.




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40. A plane angle is the difference in direction of two lines. The two lines are called the sides of the angle, and the point where the sides meet is called the vertex of the angle.

41. If there is but one angle at a
given vertex, it is designated by a cap-
ital letter placed at the vertex, and is A
read by simply naming the letter; as,
angle A (Fig. 7).

But when two or more angles have
the same vertex, each angle is desig-
nated by three letters, as shown in
Fig. 8, and is read by naming the
three letters, the one at the vertex be-
tween the others. Thus, the angle
DAC means the angle formed by the
sides AD and AC.

It is often convenient to designate an angle by placing a small italic letter between the sides and near the vertex, as in Fig. 9.

42. Two angles are equal if they can be made to coincide.


FIG. 7.

FIG. 8.



FIG. 9.


43. If the line AD (Fig. 8) is drawn so as to divide the angle BAC into two equal parts, BAD and CAD, AD is called the bisector of the angle BAC. In general, a line that divides a geometrical magnitude into two equal parts is called

a bisector of it


vertex and a common side between them; as, the angles BOD and AOD (Fig. 10).

45. When one straight line stands upon another straight line and makes the adjacent angles equal, each of these angles is called a right angle. Thus, the equal angles DCA and DCB (Fig. 11) are each a right angle.


FIG. 10.

FIG. 11.

46. When the sides of an angle extend in opposite directions, so as to be in the same straight line, the angle is straight angle. Thus, the angle formed at C(Fig its sides CA and CB extending in opposite direction is a straight angle. Hence a right angle may be half a straight angle.

47. A perpendicular to a straight line is a straight makes a right angle with it. Thus, if the angle DCA is a right angle, DC is perpendicular to AB, and A pendicular to DC.

48. The point (as C, Fig. 11) where a perpendicu another line is called the foot of the perpendicular.

49. Every angle less than a right angle is called an acute angle; as, angle A.


50. Every angle greater than a right


angle and less than a straight angle is called an oblu

as, angle C (Fig. 13).


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than two straight angles is called a reflex angle; as, angle ( (Fig. 14).


FIG. 13.



FIG 14










52. Acute, obtuse, and reflex angles, in distinction from: right and straight angles, are called oblique angles; and intersecting lines that are not perpendicular to each other are called oblique lines.


53. When two angles have the same vertex, and the sides of the one are prolongations of the sides of the other, they are called vertical angles. Thus, a and b (Fig. 15) are vertical angles.

54. Two angles are called complementary when their sum



FIG. 15

is equal to a right angle; and each is called the complement of the other; as, angles DOB and DOC' (Fig. 10).

55. Two angles are called supplementary when their sum is equal to a straight angle; and each is called the supplement of the other; as, angles DOB and DOA (Fig. 10).



56. The size of an angle depends upon the extent of opening

of its sides and not upon their length Suppose the straight

with OA, about the point O as a pivot, to the po then the line OC describes or generates the angle AOC.

The amount of rotation of the line from the position OA to the position OC is the acute angle AOC.

If the rotating line moves from the position OA to the position OB, perpendicular to OA, it generates the right angle AOB; if it moves to the position






OD, it generates the obtuse angle AOD; if it moves t tion OA', it generates the straight angle AOA'; if it the position OB', it generates the reflex angle AOB', by the dotted line; and if it continues its rotation to tion OA, whence it started, it generates two straight

Hence the whole angular magnitude about a p plane is equal to two straight angles, or four right an the angular magnitude about a point on one side of line drawn through that point is equal to one straig or two right angles.

Angles are magnitudes that can be added and su they may also be multiplied and divided by a numbe


57. If we suppose OC (Fig. 17) to turn about O from a position coincident with OA until it makes a compete revolution and comes again into D coincidence with OA, it will describe the whole angular magnitude about the point 0, while its end point C will describe a curve called a circumference.


FIG. 11

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58. By adopting a suitable unit of angles we are able to express the magnitudes of angles in numbers.

If we suppose OC (Fig. 17) to turn about O from coincidence with OA until it makes one three hundred and sixtieth of a revolution, it generates an angle at O, which is taken as the unit for measuring angles. This unit is called a degree.

The degree is subdivided into sixty equal parts called minutes, and the minute into sixty equal parts, called seconds. Degrees, minutes, and seconds are denoted by symbols. Thus, 5 degrees 13 minutes 12 seconds is written, 5° 13′ 12′′.

A right angle is generated when OC has made one-fourth of a revolution and is an angle of 90°; a straight angle is generated when OC has made one-half of a revolution and is an angle of 180°; and the whole angular magnitude about O is generated when OC has made a complete revolution, and contains 360°.

The natural angular unit is one complete revolution. But the adoption of this unit would require us to express the values of all angles by fractions. The advantage of using the degree as the unit consists in its convenient size, and in the fact that 360 is divisible by so many different integral numbers.


59. The test of the equality of two geometrical magnitudes is that they coincide throughout their whole extent.

Thus, two straight lines are equal, if they can be so placed that the points at their extremities coincide. Two angles are equal, if they can be so placed that they coincide.

In applying this test of equality, we assume that a line may be moved from one place to another without altering its length; that an angle may be taken up, turned over, and put down, without eltorin the difference in direction of its sides.

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