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a right angle. Thus, if CA, one of the lines which form the angle BAC, be extended to a point D beyond A in the same straight line, and then the angle BAD is found to be equal to the angle BAC, each of these angles is a right angle. In this case also the line BA is called a
perpendicular to the line CD; and again, AB is said to be at right angles with CD.
An obtuse angle means an angle greater than a right angle, as EAC. (8).
An acute angle means an angle less than a right angle, as EAD. (8).
11. TRIANGLES. A plane surface bounded by three straight lines meeting together at their extremities, so as entirely to enclose a space, is called a triangle; and the three straight lines are called the sides of the triangle. Thus each of the following figures is called the triangle ABC, whose sides are AB, AC, BC, the letters A, B, C, being at the three angular points.
When the three sides are equal to each other, the triangle is called equilateral, or equal-sided, as in fig. 1, where AB AC = BC*.
When two sides only are equal, as in fig. 2, where AB= AC, and BC is unequal, the triangle is called isosceles', which signifies equal-legged', as if the triangle
* The following abbreviations will be used throughout the book:for 'equals', or 'is equal to'.
+ for 'added to', or to be added.
L for 'angle'.
.. for 'therefore'.
were supposed to stand upon BC, as a base, with two legs AB, AC.
A triangle, as the name implies, has also three angles within it, as the 'angle at A', the 'angle at B', and the 'angle at C', or <BAC, ABC, and BCA: and triangles have received other distinctive names, besides those mentioned above, after the names of one or more of these angles. Thus,
A triangle, which has one of its angles a right angle, is called a right-angled triangle, as ABC fig. 3, where the angle at C' is a right angle.
A triangle, which has one of its angles an obtuse angle, is called an obtuse-angled triangle, as ABC fig. 4, where the angle at C' is an obtuse angle.
A triangle, which has each of its angles acute, is called an acute-angled triangle, as ABC figs. 1 and 2.
12. PARALLEL straight lines are such as, being in the same plane, never meet though produced ever so far both ways. Thus the straight lines
AB, CD, are parallel to each other, if, being both on the plane of this paper, they never meet however far produced either towards the right hand or the left.
13. PARALLELOGRAMS. A parallelogram is a plane surface bounded by four straight lines, called its sides, of which each opposite two are parallel.
There are several kinds of parallelograms :-viz.
(1) A square is a parallelogram, which has all its sides equal and all its angles right angles, as fig. 1.
(2) An oblong, or rectangle, is a parallelogram, which
*It will be seen hereafter that no triangle can have more than one right angle.
has all its angles right angles, but not all its sides, only the opposite ones, equal to each other, as fig. 2.
(3) A rhombus, or lozenge, is a parallelogram, which has all its sides equal, but none of its angles is a right angle, as fig. 3.
(4) A rhomboid is a parallelogram, which has its opposite sides, and not all its sides, equal; and none of its angles is a right angle, as fig. 4.
A parallelogram is generally denoted or expressed by by giving the four letters in order which are placed at the four angular points. Thus the parallelogram here traced would be called the parallelogram ABCD, or ADCB, or B BADC, &c. whichever we please.
14. A diagonal, or diameter, of a parallelogram is the straight line joining two of its opposite angular points. Thus AC, and BD are the diagonals, or diameters, of the parallelogram ABCD, in the preceding fig.
Also the side BC, upon which the parallelogram may be supposed to stand, is sometimes called its base.
15. A plane surface bounded by four straight lines of which two only are parallel, is called a trapezium, as ABCD, where AD is parallel to BC, but AB is not parallel to CD.
16. CIRCLES. A circle is a plane surface bounded by a curved line, such that every point in this line is equally distant from a certain point within the figure called the centre of the circle.
The curved boundary is called the circumference of the circle; and the straight line which measures the distance from the centre to the circumference is called the radius of the circle. Any straight line drawn through the centre and terminated both ways by the circumference is called a diameter of the circle.
Thus, in the fig. here traced, the area or surface in the
plane of the paper bounded by the curved line ABCDA is a circle, when from the centre O all straight lines to the circumference, as OA, OB, OC, OD, are equal to each other.
Any one of the lines OA, OB, OC, OD is the radius, and any radius, as AO, extended in the same straight line to meet the circumference in D, that is AD, is a diameter, of the circle.
17. Hence it is plain, that a circle may be traced by means of a string, one end of which is kept fixed in a certain point as the centre, while the other is made to revolve and trace out the circumference, the string being kept perfectly tight. The same thing is also done by the ordinary compasses.
18. A semi-circle is the half of a circle, bounded by the half of the circumference and the diameter joining its extremities.
A quadrant is the quarter of a circle, or the half of a semi-circle, bounded by the fourth-part of the circumference and two radii joining its extremities with the
Thus fig. 1 is a semicircle, and fig. 2 is a quadrant,
where is the centre of the circle in each case; and whilst ACB is half of the whole circumference in the former, it is a quarter of it in the latter.
An arc of a circle is a portion of the circumference.
It may be observed here, that although two letters are sufficient to express a straight line, three or more are generally required for a curved line; and for an obvious. reason, because between any two points there is only one straight line, but an infinite number of crooked lines, so that the extreme points entirely determine the former but not the latter.
19. It will be found, hereafter, that we often, for shortness, call the circumference of a circle the circle, which, though convenient, is not a correct way of speaking. In the same manner it is not unusual to hear persons speak of a triangle, square, or other plane surface, when, in fact, they mean no more than the boundary of the figure in each case.
Let it, then, be borne in mind, that in strictness a circle does not consist of one curved line merely, called the circumference, but that it is the whole inner area bounded by that line.
So, again, a triangle does not consist of three straight lines called sides, but is the whole inner area bounded by those sides. And similarly with respect to other plane surfaces.
20. EUCLID's Postulates' must now be admitted as truths to be granted without proof, viz.
I. A straight line may be drawn on a given plane surface from any one point to any other point.
II. A terminated straight line may be 'produced', that is, extended, to any length in a straight line.
III. A circle may be described' with any centre, and any given length, or line, for its radius.
Granted that we can do these three things, and we will assume nothing further in the construction and treatment of Geometrical figures.
In 'describing' a circle, by the third Postulate, we trace out the circumference which is the boundary of the circle. Of course we can trace a part, as well as the whole, that is, any arc of the circle.
21. EQUALITY of LINES, AREAS, and ANGLES.
It is evident that magnitudes which coincide in every part are equal to one another. This is a received axiom which admits of no dispute. It is the simplest notion we have of equality.
Hence the two straight lines AB, and CD, are equal to one another, if, when CD is placed upon AB, so that the point C is upon A and CD upon AB, the point D is found to coincide with the point B.