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in the latter which is not one of its extremities, the angles thus formed are adjacent angles.
Thus the angles ACD and BCD are adjacent angles.
g. When two straight lines cut or intersect each other, the four angles thus formed are pairs of vertically opposite angles.
Thus the angles AEC and BED are vertically opposite angles; as also are the angles AED and BEC.
When a straight line standing upon another straight line makes the adjacent angles equal to each other, each of these angles is called a RIGHT ANGLE, and each straight line is said to be PERPENDICULAR to the other.
Thus each of the angles ABD and CBD is a right angle; DB is perpendicular to AC, or to AB and BC; also AB and CB are
each perpendicular to BD.
AN OBTUSE ANGLE is one which is greater than a right angle.
AN ACUTE ANGLE is one which is less than a right angle.
A TERM OR BOUNDARY is the extremity of anything.
A FIGURE is that which is contained by one or more boundaries.
A CIRCLE is a plane figure contained by one line called the CIRCUMFERENCE, and is such that all straight lines drawn from a certain point within it, called the CENTRE, to the circumference are equal to each other. Each of such equal straight lines is called a RADIUS.
Thus BCD is a circle with circumference BCD, the centre A,
and each of the lines AB, AC, and AD is a radius. A part of the circumference, as BC, is called an arc.
Magnitude of an Angle.
1. The entire circumference of every circle is divided into 360 parts, called degrees, each deg. = 60 minutes, and each min. = 60 seconds; and an angle is said to contain as many degrees °, minutes', and seconds as are contained in the arc, or that part of the circumference which lies between the two lines forming the angle; the angular point, or vertex, being the centre of the circle.
Thus the angle BAC contains as many deg., min., and secs., as the arc BC in the smaller circle.
The angle DAE contains as many deg., min., and secs. as the arc DE in the larger circle.
Now, it can be proved that arc BC is the same part of its circumference that DE is of its circumference.
Therefore the angle BAC = the angle DAE, and hence
2. The length of the arms of an angle makes no difference in the magnitude of that angle.
Note, also, the arms need not be of the same length. For it is plain that the angle BAC = the angle BAE, &c.
3. The arms of a Right Angle include one-fourth part of the circumference. A Right Angle contains, therefore, 90 degrees; an obtuse angle contains more, and an acute angle less, than 90 degrees.
A DIAMETER OF A CIRCLE is a straight line drawn
through the centre, and terminated both ways by the circumference.
Thus BAC and DAE are diameters. A straight line drawn in a circle, not through the centre, and terminated both_ways by the circumference is called a Chord, as the straight line BE.
A SEMICIRCLE is that part of the circle which is contained by a diameter and the arc it cuts off.
In the above figure, CDB, BEC, and ECD are Semicircles.
A SEGMENT OF A CIRCLE is that part of the circle which is contained by a chord and its arc.
In the above figure the chord BE divides the circle into two segments BDCE and BFE.
RECTILINEAL FIGURES are those which are contained by right or straight lines.
A TRIANGLE is contained by three straight lines.
QUADRILATERAL FIGURES are contained by four straight
MULTILATERAL FIGURES, or POLYGONS, are contained by more than four straight lines.