A CLASSIFIED INDEX TO THE FIRST THREE BOOKS OF THE ELEMENTS OF EUCLID. THEOREMS. A. Of the Angles formed by the Meeting and Intersection of straight Lines. HYPOTHESES. If two triangles have two angles in the one respectively equal to two angles in the other, And a side of the one equal to a side of the other, either the sides adjacent to, or the sides opposite to, those equal angles. If two triangles have two sides of the one respectively equal to two sides of the other, And have also their bases equal; But if the third side of the one be greater than the third side of the other. If two triangles have two sides of the one respecively equal to two sides с he other, the angles formed by he sides also equal to one another; But if the angle formed by two sides of one be greater than the angle formed by the two sides equal to them of the other. If triangles are between the same parallels, And upon the same base, If triangles are equal in area, on the same side of it, Or upon equal bases in the same straight line, and on the same side of it. CONSEQUENCES. The remaining angles are also equal. The remaining angles and sides shall be respectively equal to one another, And the triangles themselves shall be equal to one another. The angles formed by the equal sides are equal, And the angles opposite the equal sides are equal, And the triangles themselves are equal. The angle opposite to the greater side is greater than the angle which is opposite to the less. Their bases or third sides will be equal, And the angles at the bases, which are opposite to the equal sides, will be equal, And the triangles themselves will be equal. The side opposite to that greater angle is greater than the side which is opposite to the less. They are equal to one another in area. They are between the same parallels. I. 20. D. On the Relations between the Sides and Angles of Triangles. HYPOTHESES. If any three angles are the If one angle of a triangle be Idem. If two angles of a triangle If in any triangle one angle is greater than another. If a triangle be isosceles, . CONSEQUENCES. They are together equal to two right angles. It is a right angle. beNeither of the other angles And if the equal sides be If a right-angled triangle be (Each angle at the base is isosceles. { half a right angle. The angle opposite to the greater side is greater than the angle which is opposite to the less. It is equiangular. Each angle is equal to two- The external angle is greater The external angle is equal to the sum of the two internal and opposite angles. The angle formed by the bisecting lines is equal to half the other interior and opposite angle of the triangle. The square which is constructed upon the side subtending the right angle is equal in area to the sum of the squares constructed upon the sides which form the right angle. Any figure which is constructed upon the side subtending the right angle is equal in area to the sum of the similar figures constructed upon the sides which form the right angle. I. 47, schol. 3. I. 48. II. 4, cor. 3.. HYPOTHESES. If parallelograms be constructed upon two of the sides of any triangle, and their sides parallel to the sides of the triangle be produced to meet in a point; if a straight line be drawn from that point to the vertex of the triangle, and if a parallelogram be constructed upon the base of the triangle whose other sides are equal and parallel to that straight line. If the square constructed upon one side of a triangle be equal in area to the sum of the squares constructed upon the other two sides. If from either end of the hypotenuse of a rightangled triangle parts be cut off equal to the adjacent sides. CONSEQUENCES. The last parallelogram is equal in area to the two former. The angle opposite to that side is a right angle. The square on the middle segment thus formed is equal in area to twice the rectangle under the extreme segment. I. 38, cor. II. 9, cor. 1.. I. 26, cor. 2.. II. 6, cor. 2.. E. On the Relations of Lines drawn in Triangles. |