The Elements of Solid Geometry |
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Page 5
... and PQ are parallel lines . 19. COROLLARY 2. Angles lying in different planes hav- ing their sides parallel and in the same direction each to each , are equal . We have shown that AC ' is a parallelogram ; LINES AND PLANES IN SPACE . 5.
... and PQ are parallel lines . 19. COROLLARY 2. Angles lying in different planes hav- ing their sides parallel and in the same direction each to each , are equal . We have shown that AC ' is a parallelogram ; LINES AND PLANES IN SPACE . 5.
Page 6
... sides parallel , each to each , and lying in different planes . · 20. COROLLARY 3. Lines parallel to the same line are parallel to each other . For , from the parallelograms AC ' and AB ' , we derive that , the lines CC ' and BB ' are ...
... sides parallel , each to each , and lying in different planes . · 20. COROLLARY 3. Lines parallel to the same line are parallel to each other . For , from the parallelograms AC ' and AB ' , we derive that , the lines CC ' and BB ' are ...
Page 10
... sides are in DF and DE respectively , CAB is taken as the measure of F - GD - E . Construct a second plane angle C'A'B ' , as before ; then CAB is equal to C'A'B ' . . . ( 19 ) . PROPOSITION IX . 31. THEOREM . If a line is perpendicular ...
... sides are in DF and DE respectively , CAB is taken as the measure of F - GD - E . Construct a second plane angle C'A'B ' , as before ; then CAB is equal to C'A'B ' . . . ( 19 ) . PROPOSITION IX . 31. THEOREM . If a line is perpendicular ...
Page 13
... sides AD and AB are equal . Now , AC < AB + BC ... ( 353 ) , and , AD AB , just shown . Hence , AC AD < BC , by subtraction , or , DC < BC . In the triangles DOC and BOC , since DC < BC DOC < BOC . . . ( 358 ) . angle But angle DOA ...
... sides AD and AB are equal . Now , AC < AB + BC ... ( 353 ) , and , AD AB , just shown . Hence , AC AD < BC , by subtraction , or , DC < BC . In the triangles DOC and BOC , since DC < BC DOC < BOC . . . ( 358 ) . angle But angle DOA ...
Page 17
... sides by plane or curved surfaces . A solid bounded by planes is called a polyedron . The bounding planes are called faces . The intersections of the faces are called edges . The points in which the edges meet are called vertices . 52 ...
... sides by plane or curved surfaces . A solid bounded by planes is called a polyedron . The bounding planes are called faces . The intersections of the faces are called edges . The points in which the edges meet are called vertices . 52 ...
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Common terms and phrases
ABC and DEF ABC-B altitude are equal axis base and altitude bases are equal bisecting centre circle circumference coincide common altitude common vertex conical surface COROLLARY cube cuboid DEFINITIONS diameter dicular diedral angle EDC-O equal with respect equivalent faces is called feet Find the volume frustum Hence inscribed prism intersection lateral edges lateral faces lateral surface Let the line line BA lune MN and PQ mutually equal O-ABCD oblique parallelopiped parallel planes parallelogram pass perimeter perpen perpendicular to MN plane angle plane MN plane PQ polar triangle polyedral angle prisms whose bases radii radius regular polyedrons regular polygon right angles right circular cone right prism right section SCHOLIUM sides slant height sphere spherical angle spherical excess spherical polygon spherical triangle straight line tetraedron THEOREM triangles ABC triangular prism triangular pyramids triedral
Popular passages
Page 48 - Assuming that the areas of two triangles which have an angle of the one equal to an angle of the other are to each other as the products of the sides including the equal angles...
Page 46 - Similar cylinders are to each other as the cubes of their altitudes, or as the cubes of the diameters of their bases.
Page 4 - If a straight line is perpendicular to each of two other straight lines at their point of intersection, it is perpendicular to the plane of the two lines.
Page 49 - A sphere is a solid bounded by a curved surface, every point of which is equally distant from a point within called the center.
Page 94 - Two triangles are equal if two sides and the included angle of the one are equal respectively to two sides and the included angle of the other (sas = sas). Hyp. In A ABC and A'B'C', AB = A'B', BC = B'C', and Z B = Z B'.
Page 46 - The lateral areas, or the total -areas, of two similar cones of revolution are to each other as the squares of their altitudes...
Page 6 - Theorem: If a straight line is perpendicular to one of two parallel planes, it is perpendicular, also, to the other plane.
Page 9 - ... meeting the plane at unequal distances from the foot of the perpendicular the more remote is the greater.
Page 56 - A spherical angle is measured by the arc of a great circle described from its vertex as a pole, and included between its sides, produced if necessary.