An Elementary Treatise on Plane and Solid Geometry |
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Page 15
... triangles are equal , when two sides and the included angle of the one are respective- ly equal to two sides and the included angle of the other . Demonstration . In the two triangles ABC , DEF ( fig . 31 ) , let the angle A be equal to ...
... triangles are equal , when two sides and the included angle of the one are respective- ly equal to two sides and the included angle of the other . Demonstration . In the two triangles ABC , DEF ( fig . 31 ) , let the angle A be equal to ...
Page 16
... triangles are equal , when a side and the two adjacent angles of one triangle are respectively equal to those of the other . Demonstration . In the two triangles ABC , DEF ( fig . 31 ) , let the side AB be equal to the side DE , and the ...
... triangles are equal , when a side and the two adjacent angles of one triangle are respectively equal to those of the other . Demonstration . In the two triangles ABC , DEF ( fig . 31 ) , let the side AB be equal to the side DE , and the ...
Page 17
Benjamin Peirce. Equal Angles of the Isosceles Triangle . Demonstration . In the isosceles triangle ABC ( fig . 32 ) , let the equal sides be AB and BC . Let the line BD be drawn so as to bisect the angle ABC . Then the two triangles ADB ...
Benjamin Peirce. Equal Angles of the Isosceles Triangle . Demonstration . In the isosceles triangle ABC ( fig . 32 ) , let the equal sides be AB and BC . Let the line BD be drawn so as to bisect the angle ABC . Then the two triangles ADB ...
Page 18
Benjamin Peirce. Third Case of Equal Triangles . 59. Corollary . An equiangular triangle is also equi- lateral . 60 ... ABC and DEF ( fig . 31 ) be the triangles , whose sides AB , BC , and AC are , respectively equal to DE , EF , and DF ...
Benjamin Peirce. Third Case of Equal Triangles . 59. Corollary . An equiangular triangle is also equi- lateral . 60 ... ABC and DEF ( fig . 31 ) be the triangles , whose sides AB , BC , and AC are , respectively equal to DE , EF , and DF ...
Page 19
... triangle CAG is thus formed , and AB being per- pendicular to its base , divides it , by art . 57 , into the two equal triangles ABC and ABG . 64. Theorem . The sum of the three angles of any triangle is equal to two right angles ...
... triangle CAG is thus formed , and AB being per- pendicular to its base , divides it , by art . 57 , into the two equal triangles ABC and ABG . 64. Theorem . The sum of the three angles of any triangle is equal to two right angles ...
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Common terms and phrases
ABCD &c AC AC adjacent angles altitude angle ABC angle BAC arc BC base and altitude bisect CD fig centre chord circumference convex surface Corollary cylinder DEF fig Definitions Demonstration denote diameter divided Draw equal arcs equal distances equilateral equivalent four right angles frustum given angle given circle given line given polygon given sides given square greater half the product Hence homologous sides hypothenuse infinitely small inscribed circle isoperimetrical isosceles Join AC Let ABCD line AB fig lines drawn mean proportional number of sides oblique lines parallel lines parallelogram parallelopiped perimeter perpendicular plane angles plane MN polygon ABCD prism Problem radii radius ratio rectangles regular polygon respectively equal right triangles Scholium secant sector segment side AC similar polygons similar triangles solid angle Solution sphere spherical polygon spherical triangle tangent Theorem triangles ABC triangular prism vertex vertices whence
Popular passages
Page 148 - 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. D c A' D' Hyp. In triangles ABC and A'B'C', ZA = ZA'. To prove AABC = ABxAC. A A'B'C' A'B'xA'C' Proof. Draw the altitudes BD and B'D'.
Page 90 - To construct a parallelogram equivalent to a given square, and having the difference of its base and altitude equal to a given line.
Page 24 - CIRCLE is a plane figure bounded by a curved line, all the points of which are equally distant from a point within called the centre; as the figure ADB E.
Page 40 - One side and two angles of a triangle being given, to construct the triangle. Solution.
Page 79 - Construct, by § 145, a right triangle, of which the hypothenuse BC (fig. 79) is equal to the side of the greater square, and the leg AB is equal to the side of the less square ; and AC is the side of the required square.
Page 142 - THEOREM. If two triangles on the same sphere, or on equal spheres, are mutually equiangular, they will also be mutually equilateral. Let A and B be the two given triangles; P and Q their polar triangles. Since the angles are equal in the triangles A and B, the sides will be equal in. their polar triangles P and Q (Prop.
Page 6 - The preface and commentary to the Antigone are even more creditable to Mr. Woolsey's ability than those to the Alcestis. The sketch of the poem, in the preface, is written with clearness and brevity. The difficulties in this play, that call for a commentator's explanation, are far more numerous than in the Alcestis.
Page 70 - The perimeters of two regular polygons of the same number of sides, are to each other as their homologous sides, and their areas are to each other as the squares of those sides (Prop.
Page 5 - A surface is that which has length and breadth, without thickness. 6. A plane is a surface, in which any two points being taken, the straight line joining those points lies wholly in that surface.
Page 137 - Each side of a spherical triangle is less than the sum of 'the other two sides. 48. The sum of the sides of a spherical polygon is less than 360°.