Plane and Solid Geometry |
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Page 9
... Proof . Apply ≤ DEF to ≤ ABC so that the vertex E coin- cides with the vertex B , and ED coincides with BA . Then EF will fall on BC ( straight lines coinciding in part coincide throughout ) . Hence DEF = LABC . 53. All right angles ...
... Proof . Apply ≤ DEF to ≤ ABC so that the vertex E coin- cides with the vertex B , and ED coincides with BA . Then EF will fall on BC ( straight lines coinciding in part coincide throughout ) . Hence DEF = LABC . 53. All right angles ...
Page 10
... Proof . 21 = 22 . 21 is a supplement of 3 , 2 is a supplement of 3 , ( two adjacent angles whose exterior sides are in a straight line are supplementary ) . .. 21 = 22 , ( supplements of equal △ are equal ) . Q.E.D. Ex . 26. If , in ...
... Proof . 21 = 22 . 21 is a supplement of 3 , 2 is a supplement of 3 , ( two adjacent angles whose exterior sides are in a straight line are supplementary ) . .. 21 = 22 , ( supplements of equal △ are equal ) . Q.E.D. Ex . 26. If , in ...
Page 12
... ABC and A'B'C ' , AB = A'B ' , ZA ZA ' , and B = B ' . To prove = △ ABC = △ A'B'C ' . Proof . Apply △ ABC to △ A'B'C ' so that AB shall coin- cide with A'B ' . BC will take the direction of B'C ' , ( 12 PLANE GEOMETRY.
... ABC and A'B'C ' , AB = A'B ' , ZA ZA ' , and B = B ' . To prove = △ ABC = △ A'B'C ' . Proof . Apply △ ABC to △ A'B'C ' so that AB shall coin- cide with A'B ' . BC will take the direction of B'C ' , ( 12 PLANE GEOMETRY.
Page 15
... A = 2 C. PROPOSITION IV . THEOREM 75. An exterior angle of a triangle is greater than either remote interior angle . Hyp . To prove а BCD is an ext . Z of △ ABC . 4 BCD > < A or △ B. Proof . Let E be the midpoint of BC . TRIANGLES 15.
... A = 2 C. PROPOSITION IV . THEOREM 75. An exterior angle of a triangle is greater than either remote interior angle . Hyp . To prove а BCD is an ext . Z of △ ABC . 4 BCD > < A or △ B. Proof . Let E be the midpoint of BC . TRIANGLES 15.
Page 18
... Proof . AC and DF either meet or are parallel . Suppose they meet in G. Then BEG is a triangle whose exterior ABE is equal to a remote interior / BEG , which is impossible . Hence AC and DF are parallel . Q.E.D. 81. SCHOLIUM . That the ...
... Proof . AC and DF either meet or are parallel . Suppose they meet in G. Then BEG is a triangle whose exterior ABE is equal to a remote interior / BEG , which is impossible . Hence AC and DF are parallel . Q.E.D. 81. SCHOLIUM . That the ...
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Common terms and phrases
ABCD altitude angles are equal bisect bisector chord circumference circumscribed cone construct a triangle cylinder diagonals diagram for Prop diameter diedral angles divide draw drawn equiangular equiangular polygon equilateral triangle equivalent exterior angle face angles find a point Find the area Find the radius Find the volume frustum given circle given line given point given triangle Hence HINT homologous hypotenuse inches inscribed intersecting isosceles triangle joining the midpoints lateral area lateral edges line joining mean proportional median opposite sides parallel lines parallelogram parallelopiped perimeter perpendicular plane MN point equidistant polyedral angle polyedron PROPOSITION prove Proof quadrilateral radii ratio rectangle regular polygon respectively equal rhombus right angles right triangle SCHOLIUM segments sphere spherical polygon spherical triangle square straight angle straight line surface tangent THEOREM trapezoid triangle ABC triangle are equal triedral vertex
Popular passages
Page 148 - If two chords intersect within a circle, the product of the segments of one is equal to the product of the segments of the other.
Page 150 - If, from a point without a circle, a tangent and a secant be drawn, the tangent is the mean proportional between the secant and its external segment.
Page 180 - Two triangles having 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 45 - If two triangles have two sides of one equal respectively to two sides of the other, but the included angle of the first triangle greater than the included angle of the second, then the third side of the first is greater than the third side of the second.
Page 337 - A spherical polygon is a portion of the surface of a sphere bounded by three or more arcs of great circles. The bounding arcs are the sides of the polygon ; the...
Page 305 - A cylinder is a solid bounded by a cylindrical surface and two parallel planes ; the bases of a cylinder are the parallel planes; and the lateral surface is the cylindrical surface.
Page 312 - The lateral areas, or the total areas, of similar cylinders of revolution are to each other as the squares of their altitudes, or as the squares of their radii ; and their volumes are to each other as the cubes of their altitudes, or as the cubes of their radii. Let S, S' denote the lateral areas, T, T...
Page 149 - If, from a point without a circle, two secants are drawn, the product of one secant and its external segment is equal to the product of the other and its external segment.
Page 328 - Every section of a sphere made by a plane is a circle.
Page 257 - If two intersecting planes are each perpendicular to a third plane, their intersection is perpendicular to that plane.