PLANIMETRY. o 
of a quadrilateral are together equal to four right angles, as will become 
evident if we divide the quadrilateral by a diagonal into two triangles (pl. 1, 
fig. 23). The sum of the angles in a pentagon is equal to six right angles, 
since two diagonals divide it into three triangles (pl. 3, fig.4). And in general, 
the sum of the angles of a rectilineal figure is always equal to twice as 
many right angles, less 4, as the figure has sides. This proposition 
will become clearer if we draw, from a given point within the figure, lines 
to all its corners (fig. 5), and remember that the sum of all angles that have 
their vertex in one common point, is equal to four right angles. This 
proposition also holds good if the figure have a re-entrant angle (fig. 6) ; but 
in order to prove it in that case, it will be better to divide the figure 
by diagonals that must not intersect one another, into triangles, of 
which there will always be two less than the figure has sides. 
3. OF THE EQUALITY OF FIGURES. 
Two figures are said to be equal wnen they can be so applied to each 
other as to coincide throughout. The sides and angles of a figure are in 
such intimate and dependent relation, that from the equality of some 
of them we may infer the equality of the rest. For example, if of two 
triangles we know that three parts are mutually equal—either the three 
sides, or two sides and the included angle, or two sides and the angle which 
is opposite the greater of the two, or two angles and the included side— 
then may we conclude from this that the rest of the parts are also equal each 
to each, and that the triangles themselves are equal (pl. 3, figs. 11, 12). 
But of the three parts ascertained, one must always be a side, since 
two triangles of unequal sides may have equal angles, as in fig. 7. If, in 
this triangle ade, we add to de the parallel bc, then it is plain that the 
triangles abe and ade have their angles equal, since g = ¢,m = n,o =p; but 
the triangles themselves are by no means equal, since the one is only a part 
of the other. From these cases of equality it follows also what parts are 
necessary to construct a triangle. This is most easily done by having the 
three sides, a, b, c, given (fig. 8) ; but we may also employ two sides, a, b, and 
the included angle m (fig. 9), as well as two angles, m,n, and the included 
side a (fig. 10); or finally, two sides, a, b, and the angle m (fig. 18) lying 
opposite to one of them. It is to be observed, however, that when this 
angle lies opposite to the smaller of the two sides, two different triangles 
may be constructed, both of which will answer the conditions of the propo- 
sition, so that in this case the triangle is not completely defined. By means 
of the equality of triangles the following, among other properties, may also 
be proved: 1. In an isosceles triangle, the angles opposite the equal sides 
are also equal (fig. 13), for let ab = ac, and from a@ draw a line which 
bisects bc, then there will be two equal triangles in which Zm= Zn, from 
which it follows that o= p, and g=,r, which shows that the line is perpen- 
dicular to 6c, and bisects the angle at a@. In an equilateral triangle, all the 
three angles are equal. 2. The greater angle of a triangle lies opposite to 
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