An Elementary Treatise on Plane and Solid GeometryJ. Munroe, 1837 - 159 σελίδες |
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Σελίδα 7
... = MPC + MPD two right angles . b . In the same way , APC and BPC may be proved to be supplements of each other ; and therefore APD and Adjacent and Vertical Angles . Sum of all the Angles CH . IV . § 23. ] 7 THE ANGLE .
... = MPC + MPD two right angles . b . In the same way , APC and BPC may be proved to be supplements of each other ; and therefore APD and Adjacent and Vertical Angles . Sum of all the Angles CH . IV . § 23. ] 7 THE ANGLE .
Σελίδα 10
... proved equal to MND . 31. Theorem . If two straight lines , lying in the same plane , as AB , CD ( fig . 13 ) , are cut by a third , EF , so that the angles EMB and END are equal , or AMN and MND are equal , & c .; the lines AB , CD ...
... proved equal to MND . 31. Theorem . If two straight lines , lying in the same plane , as AB , CD ( fig . 13 ) , are cut by a third , EF , so that the angles EMB and END are equal , or AMN and MND are equal , & c .; the lines AB , CD ...
Σελίδα 22
... proved that BAD = BCD . In the 78. Corollary . Two parallel lines comprehended between two other parallel lines are equal . 79. Theorem . If , in a quadrilateral ABCD ( fig . 39 ) , the opposite sides are equal , namely , AB = CD , and ...
... proved that BAD = BCD . In the 78. Corollary . Two parallel lines comprehended between two other parallel lines are equal . 79. Theorem . If , in a quadrilateral ABCD ( fig . 39 ) , the opposite sides are equal , namely , AB = CD , and ...
Σελίδα 36
... prove that the line ABM , perpendicular to the common tangent at M , passes through both the centres A and B. CHAPTER IX . PROBLEMS RELATING TO THE FIRST EIGHT CHAPTERS . 128. Problem . To find the position of a point in a plane ...
... prove that the line ABM , perpendicular to the common tangent at M , passes through both the centres A and B. CHAPTER IX . PROBLEMS RELATING TO THE FIRST EIGHT CHAPTERS . 128. Problem . To find the position of a point in a plane ...
Σελίδα 43
... proved that OF OD = OE . = Hence the circumference DFE passes through the points D , F , E , and the sides are tangents to it , by art . 120 . 152. Corollary . The three lines AO , BO , and CO , which bisect the three angles of a ...
... proved that OF OD = OE . = Hence the circumference DFE passes through the points D , F , E , and the sides are tangents to it , by art . 120 . 152. Corollary . The three lines AO , BO , and CO , which bisect the three angles of a ...
Άλλες εκδόσεις - Προβολή όλων
Συχνά εμφανιζόμενοι όροι και φράσεις
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
Δημοφιλή αποσπάσματα
Σελίδα 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'.
Σελίδα 90 - To construct a parallelogram equivalent to a given square, and having the difference of its base and altitude equal to a given line.
Σελίδα 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.
Σελίδα 40 - One side and two angles of a triangle being given, to construct the triangle. Solution.
Σελίδα 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.
Σελίδα 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.
Σελίδα 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.
Σελίδα 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.
Σελίδα 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.
Σελίδα 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°.