PROPOSITION 48. THEOREM. If the square described on one of the sides of a triangle be equal to the squares described on the other two sides, the angle contained by these two sides is a right angle. Let the sq. described on BC, one of the sides of ▲ ABC, be equal to the sqs. described on the other two sides BA, AC, then BAC shall be a rt. angle. and AC is common to the two As BAC and DAC, [CONST. NOTES. We have here a converse proved directly. Ex. 97.-Squares on equal straight lines are equal. (Superposition.) Ex. 98. The square on the greater of two unequal straight lines is greater than the square on the less. Ex. 99.-Equal squares are on equal straight lines. reference from 98.) (Immediate Ex. 100. Two congruent rectangles can be placed, with the square on the difference of their sides, so as to make up the squares on the two sides. Ex. 101. Two congruent rectangles, together with the squares on their sides, can be placed so as to make the square on the sum of the two sides. Ex. 102.-If we read 'greater than' for 'equal to' in the Enunciation of I. 48, we must read 'obtuse' for 'right.' Ex. 103.-Make corresponding substitution for 'less,' 'acute.' Ex. 104.-State and prove the converses of 102, 103. DEFINITIONS, POSTULATES, AND AXIOMS. For the sake of reference, and for examination purposes, a complete set of Definitions, Postulates, and Axioms is appended. DEFINITIONS. 1. A Point is that which has no parts, or which has no magnitude. 2. A Line is length without breadth. 3. The Extremities of a Line are points. 4. A Straight Line is that which lies evenly between its extreme points. 5. A Superficies is that which has only length and breadth. 6. The Extremities of a Superficies are lines. 7. A Plane Superficies is that in which any two points being taken, the straight line between them lies wholly in that superficies. 8. A Plane Angle is the inclination of two lines to one another in a plane, which meet together, but are not in the same direction. 9. A Plane Rectilineal Angle is the inclination of two straight lines to one another, which meet together, but are not in the same straight line. 10. When a straight line standing on another straight line makes the adjacent angles equal to one another, each of the angles is called a Right Angle; and the straight line which stands on the other is called a Perpendicular to it. II. An Obtuse Angle is that which is greater than a right angle. 12. An Acute Angle is that which is less than a right angle. 13. A Term or Boundary is the extremity of anything. 14. A Figure is that which is inclosed by one or more boundaries. 15. A Circle is a plane figure contained by one line, which is called the circumference, and is such that all straight lines drawn from a certain point within the figure to the circumference are equal to one another. 16. And this point is called the Centre of the circle. 17. A Diameter of a circle is a straight line drawn through the centre, and terminated both ways by the circumference. 18. A Semicircle is the figure contained by a diameter and the part of the tircumference cut off by the diameter. 19. A Segment of a circle is the figure contained by a straight line and the circumference it cuts off. 20. Rectilineal Figures are those which are contained by straight lines. 21. Trilateral Figures, or Triangles, by three straight lines. 22. Quadrilateral, by four straight lines. 23. Multilateral Figures, or Polygons, by more than four straight lines. 24. Of three-sided figures, an Equilateral Triangle is that which has three equal sides. 25. An Isosceles Triangle is that which has only two sides equal. 27. A Right-Angled Triangle is that which has a right angle. 30. Of four-sided figures, a Square is that which has all its sides equal, and all its angles right angles. 31. An Oblong is that which has all its angles right angles, but has not all its sides equal. 32. A Rhombus is that which has all its sides equal, but its angles are not right angles. 33. A Rhomboid is that which has its opposite sides equal to one another, but all its sides are not equal, nor its angles right angles. 34. All other four-sided figures besides these are called Trapeziums. 35. Parallel Straight Lines are such as are in the same plane, and which, being produced ever so far both ways, do not meet. 36. A Parallelogram is a four-sided figure whose opposite sides are parallel, and the Diameter (or Diagonal) is the straight line joining two of its opposite angles. Let it be granted : POSTULATES.1 1. That a Straight Line may be drawn from any one point to any other point. 2. That a Terminated Straight Line may be produced to any length in a straight line. 3. And that a Circle may be described from any centre, at any distance from that centre. 1 i.e. Elementary Problems whose construction it is to be taken for granted we can effect. AXIOMS.1 1. Things which are equal to the Same are Equal to One Another. 2. If equals be Added to equals, the Wholes are equal. 3. If equals be Taken from equals, the Remainders are equal. 4. If equals be Added to unequals, the Wholes are unequal. 5. If equals be Taken from unequals, the Remainders are unequal. 6. Things which are Double of the same are Equal to one another. 7. Things which are Halves of the same are Equal to one another. 8. Magnitudes which Coincide with one another, that is, which exactly fill the same space, are Equal to one another. 9. The Whole is greater than its Part. 10. Two Straight Lines cannot inclose a Space. II. All Right Angles are equal to one another. 12. 'If a straight line meets two straight lines, so as to make the two interior angles on the same side of it taken together less than two right angles, these straight lines being continually produced, shall at length meet upon that side on which are the angles which are less than two right angles.' 1i.e. Elementary Theorems whose truth is taken for granted. |