circumference of the circle, and the angle between them is known as an angle of one degree. If, in the circle, two radii are drawn perpendicular to each other, they enclose a quarter of the circle, and hence a right angle consists of ninety degrees, written 90°. Each degree is divided into 60 equal parts, or minutes; and each minute is again subdivided into 60 equal parts called seconds.

Abbreviations are used for these denominations. Thus, 52° 14′ 20.5′′ denotes 52 degrees 14 minutes 20.5 seconds.

The following definitions will often be required.

A triangle is a plane figure bounded by three straight lines; any one of the three angular points A, B, or C, Fig. 10, may be looked upon as the vertex, the opposite side is then called the base of the triangle. The altitude of a triangle is the perpendicular distance of the vertex from the base.

B

B

FIG. 10.-A triangle.

Equilateral triangle. When the three sides of a triangle are equal, the triangle is an equilateral triangle, also the angles of the triangle are equal, each being 60°.

Isosceles triangle.-When two sides of a triangle are equal, the triangle is an isosceles triangle.

A right-angled triangle, Fig. 11, is a triangle one angle C of which is a right angle; the side AB opposite the right angle is called the hypotenuse.

A plane angle is the inclination of two lines which meet each other but are not in the same straight line. Thus, if a line AO meet a line OC at 0, Fig. 12, the amount of opening between the lines CO, OA is called the angle AOC. If only one angle is formed at 0, the angle may be written as the angle 0;

but if several angles come together at the same point, the middle letter indicates the vertex of the angle referred to. The magnitude of the angle in degrees, minutes and seconds may be indicated as shown

in the angle COA Fig. 12, or Greek letters may be used for the same purpose. The

angle COA may be designated by 0, the angle AOD by 4, and the angle DOE by a. It must be very carefully observed that the angle is independent of the length of the lines forming the sides

E

or legs of the angle. Thus the angle may be accurately described either as the angle AOC, or MON.

Representation and Measurement of Angles.--As the length of the lines forming the two sides of an angle have no connection with the magnitude of the angle, the actual size is best expressed by the fraction of a circle which the angle in question subtends at its centre. This is done in the following

manner.

With centre 0 and any convenient radius, describe a circle CBDE, as shown in Fig. 13.

If we suppose a small pointer (such as the minute hand of a clock or watch) free to move about the centre O, made to coincide with OC and afterwards made to move from C towards B to a position A, through an arc CA one-sixth of the circumference, then the angle COA is

a sixth part of 360°, or is an angle of 60 degrees, written as 60°. In a similar manner, if it moves to B it will trace out an angle of 90 degrees.

When it moves to a position A' it is evident that the angle traced out is greater than a right angle. All angles greater than a right angle are called obtuse angles. Consequently, the angle COA' is an obtuse angle. Angles less than 90°, or less than a right angle, are acute. The angle COA is an acute angle. From the foregoing considerations, it will be seen that an angle is measured by the number of degrees in the arc of the circle, having the vertex of the angle as its centre, intercepted by the two lines forming the angle.

Comparison of the Magnitudes of Angles.-A comparison of the magnitudes of two angles ABC and DEF (Fig. 14) may be

C

C

F

B

A E

FIG. 14.-Comparison of the magnitudes of two angles.

made by placing the angle DEF on the angle ABC, so that the point E exactly falls upon the point B, and the line DE along the line AB, then if the line EF falls on the line BC the angles are said to be equal. The angle DEF is less than the angle ABC if EF falls within BC, as shown by the dotted line BC'. It is larger if it falls outside BC. This method of superposition is readily performed in the following way: Draw from centres B and E, as shown, two equal arcs AC and DF, so that DE and EF in the one case are equal to AB and BC respectively in the other. If the point A be joined to the point C, and D to F, then, if AC is equal to DF, the triangles ABC and DEF are obviously equal; or, using a piece of tracing paper, make a tracing of DEF, and placing the tracing on ABC, the comparison is readily made.

To set out a given angle.---By means of what is called a 60° set square shown at (i) (Fig. 15), angles of 60°, 30°, 90° and 120° can be set out. Also by means of the 45° set square shown at

(ii), angles of 45°, 90°, and 135° can be marked off. Other angles, viz. 15°, 22°, and 75° can also be obtained by using these set squares and a pair of compasses.

45

FIG. 15 (ii).-A 45° set-square.

60

FIG. 15 (i).-A 60° set-square.

C

D

To set out an angle of 15°.-Make an angle ABC (Fig. 16) equal to 30° with the set square (i). Bisect the angle ABC by a line BD, then ABD and DBC are each 15°. To bisect the angle ABC we proceed as follows:With B as centre, and any convenient radius BA, describe an arc AC. With A as centre and any radius describe an arc, and from C, using the same

B

15°

A

FIG 16. To set out an angle of 15°.

radius, describe another arc cutting the former in point D. Join

D to B, then BD bisects the

angle ABC, and therefore makes an angle of 15° with AB.

An angle of 221⁄2° is obtained in a similar manner. Make an angle ABC (Fig. 17) equal to 45° with the set square (ii). Bisect the angle ABC by the line BD as described

above; then the angle ABD is equal to 22°.

An angle of 75°.-To obtain an angle of 75° it is only necessary to use the two set squares. Thus the angle ABC is made equal to 45° by using set square (ii), next adding to this the angle 30° by means of set square (i). The angle ABE=75°.

If the 60° angle of set square (i) be added to the angle ABC (Fig. 17) the angle obtained will be 105°.

Use of Protractor.--Angles which are not conveniently obtained by construction are set out by means of a protractor. Two forms of such protractors are shown in Fig. 18. The first

consists of a thin flat rule or scale made of boxwood, ivory, or other material, along the edge of which angles are marked. These marks are obtained from the corresponding division on a semicircle as

shown in the illus

tration.

Ex. 1. Set out by a protractor an angle of 50°.

At the point P (Fig. 19) we place

the mark of Fig..

18 coincident with P

N

M

P, and the edge of FIG. 19.-To set out an angle by means of a protractor. the protractor BC with the line PM.