Plane and Solid Geometry |
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Page 3
... prove the truth of a theorem , or to solve a problem . 25. The Enunciation of a theorem consists of two parts : the Hypothesis , or that which is assumed ; and the Conclusion , or that which is asserted to follow therefrom . POSTULATES ...
... prove the truth of a theorem , or to solve a problem . 25. The Enunciation of a theorem consists of two parts : the Hypothesis , or that which is assumed ; and the Conclusion , or that which is asserted to follow therefrom . POSTULATES ...
Page 9
... prove that ACB is a straight line . If ACB is not a straight line , let CE be in the same straight line with AC . * Hereafter converse propositions will not be demonstrated , but given as corollaries or exercises . The method of ...
... prove that ACB is a straight line . If ACB is not a straight line , let CE be in the same straight line with AC . * Hereafter converse propositions will not be demonstrated , but given as corollaries or exercises . The method of ...
Page 10
... prove that By ( 44 ) , and Therefore , by ( 28 ) , Subtracting we have = ZAEC + 2 CEB = △ BED + ≤ CEB . Z CEB CEB , = LAEC = ZBED . In the same way it may be proved that LAED CEB . = Q.E.D. 50. COR . 1. If two straight lines intersect ...
... prove that By ( 44 ) , and Therefore , by ( 28 ) , Subtracting we have = ZAEC + 2 CEB = △ BED + ≤ CEB . Z CEB CEB , = LAEC = ZBED . In the same way it may be proved that LAED CEB . = Q.E.D. 50. COR . 1. If two straight lines intersect ...
Page 12
... prove that PO is the only perpendicular that can be drawn , and that it is the shortest distance from P to AB . Produce PO to P ' , making OP ' = OP ; then the angles POB and P'OB are right angles . If any other perpendicular can be ...
... prove that PO is the only perpendicular that can be drawn , and that it is the shortest distance from P to AB . Produce PO to P ' , making OP ' = OP ; then the angles POB and P'OB are right angles . If any other perpendicular can be ...
Page 13
... prove that CA = CB . Let the part CDA be revolved about CD ; then , since AD = DB by hypothesis , and CD remains stationary , the point A will fall on B , and the line CA will fall on CB , and be equal to it . Hence CA = CB . Q.E.D. 54 ...
... prove that CA = CB . Let the part CDA be revolved about CD ; then , since AD = DB by hypothesis , and CD remains stationary , the point A will fall on B , and the line CA will fall on CB , and be equal to it . Hence CA = CB . Q.E.D. 54 ...
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Common terms and phrases
ABCD AC² acute angle AD² adjacent adjacent angles altitude angle formed angles are equal apothem arc BC base and altitude bisect bisector called centre chord circumference circumscribed cone cylinder diagonals diameter diedral angles distance divided draw drawn ECDH equally distant equilateral equivalent EXERCISES faces four right angles frustum given point given straight line hence homologous homologous sides hypotenuse inscribed polygon interior angles intersection isosceles triangle join lateral area lateral edges Let ABC lune mean proportional measured by one-half middle point number of sides parallelogram parallelopiped perimeter perpendicular polyedral angle polyedron PROPOSITION XI prove pyramid Q.E.D. PROPOSITION quadrilateral radii radius ratio rectangle rectangular parallelopiped regular polygon right triangle SCHOLIUM segments semiperimeter sphere spherical angle spherical polygon spherical triangle surface tangent THEOREM triangle ABC triangles are equal triangular triangular prism V-ABC vertex vertical angle
Popular passages
Page 46 - PERIPHERY of a circle is its entire bounding line ; or it is a curved line, all points of which are equally distant from a point within called the centre.
Page 105 - ... any two parallelograms are to each other as the products of their bases by their altitudes. PROPOSITION V. THEOREM. 403. The area of a triangle is equal to half the product of its base by its altitude.
Page 82 - If any number of quantities are proportional, any antecedent is to its consequent as the sum of all the antecedents is to the sum of all the consequents. Let a : b = c : d = e :f Now ab = ab (1) and by Theorem I.
Page 192 - A sphere is a solid bounded by a surface all points of which are equally distant from a point within called the centre.
Page 108 - 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 146 - A STRAIGHT line is perpendicular to a plane, when it is perpendicular to every straight line which it meets in that plane.
Page 30 - In an isosceles triangle, the angles opposite the equal sides are equal.
Page 80 - In any proportion the terms are in proportion by Composition ; that is, the sum of the first two terms is to the first term as the sum of the last two terms is to the third term.
Page 79 - If the product of two quantities is equal to the product of two others, one pair may be made the extremes, and the other pair the means, of a proportion. Let ad = ос.
Page 148 - Equal oblique lines from a point to a plane meet the plane at equal distances from the foot of the perpendicular ; and of two unequal oblique lines the greater meets the plane at the greater distance from the foot of the perpendicular.