STEREOMETRY. i 13 
II. STEREOMETRY, OR THE GEOMETRY OF SOLIDS. 
1. OF THE POSITION OF LINES AND PLANES IN SPACE. 
Through one or two points, as well as through one straight line, innu- 
merable planes may pass. Only one, however, can pass through three 
points not in the same straight line, through two parallel or intersecting 
straight lines, or through a straight line and a point external to it. Two 
planes meeting each other without coinciding, form a straight line by their 
intersection. A straight line not in a plane, can have only one point in 
common with it. It is perpendicular to the plane when it is perpendicular 
to all straight lines drawn through its foot in the plane; this is likewise the 
case when it forms a right angle with two lines lying in the plane (pil. 3, 
jigs. 86, 87). The angle of inclination of a line to a plane not perpen- 
dicular to it, is found by letting fall a perpendicular from any point of the 
line upon the plane, and connecting the extremities of the two lines by a 
line situated in the plane. This is the least angle which the straight line 
can make with lines drawn through its footon the plane. Two straight 
lines perpendicular to the same plane, are parallel to each other (fig. 90). 
If, of two parallels, one stands perpendicular to a plane, the other must also. 
A straight line is parallel to a plane, as well as a plane parallel to a straight 
line, when they will not meet if produced. If a straight line be parallel to 
a plane, and we pass through the line, planes cutting the first plane, the 
lines of intersection will be parallel to each other and to the plane (fig. 88). 
Two planes perpendicular to the same straight line are parallel to each 
other (fig. 90). Two parallel planes, intersected by a third, will have the 
lines of intersection parallel (fig. 91). If two straight lines in space be 
intersected by three parallel] planes, the segments of the one will be pro- 
portional to those of the other (jig. 93). Parallels or perpendiculars 
between two parallel planes are equal; hence the distance between two 
parallel planes is measured by a perpendicular let fall from one upon the 
other. ‘T'wo angles which have three sides parallel are equal; if they lie in 
different planes, these latter are parallel (fig. 92). The inclination or 
separaticn of two planes not parallel, is measured by the angle formed by 
lines in each plane, drawn perpendicular to a point in the line of inter- 
section of the planes. Planes, like lines, may form adjacent and opposite 
or vertical angles, which, with respect to their magnitudes, have the same 
properties as those of lines (fig. 89). Two planes are perpendicular when 
their angle is aright angle. Ifa plane be perpendicular to two intersecting 
planes, it will also be perpendicular to their line of intersection. If a 
straight line be perpendicular to a plane, every plane passing through the 
former will be perpendicular to the latter. When three or more planes 
meet in one point, they form a corner or solid angle (pl. 3, fig. 94). The 
edges or lines of intersection of planes meeting in this manner, form as 
many plane angles as there are planes. If the solid angle be formed by 
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