I. 32B, cor. 3. I. 26. I. 8, and cor. I. 25. I. 4. I. 24. I. 37. I. 38. I. 39. I. 40. 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 respectively equal to two sides of the other, And the angles formed by those 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, Or upon equal bases. If triangles are equal in area, And upon the same base, and 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. D. On the Relations between the Sides and Angles of Triangles. HYPOTHESES. If any three angles are the internal angles of a triangle. If one angle of a triangle be equal to the other two. If one angle of a triangle be a right angle. Idem. If two angles of a triangle are equal. If in any triangle one angle is greater than another. If a triangle be isosceles. And if the equal sides be produced. CONSEQUENCES. They are together equal to two right angles. It is a right angle. Neither of the other angles can be a right angle. The other two are together equal to a right angle. The sides opposite to those angles are also equal. The side which is opposite to the greater angle is greater than the side which is opposite to the less. The angles at the base are equal to one another. The angles formed by the produced sides and the base, below the same, shall be equal. If a right-angled triangle be Each angle at the base is isosceles. If one side of any triangle be greater than another. If a triangle is equilateral. If a triangle is equiangular.. If one side of a triangle be produced. I. 32 A. Idem. 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 equal to twothirds of a right angle. It is equilateral. The external angle is greater than either of the internal opposite angles. 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. L47 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. E. On the Relations of Lines drawn in Triangles. HYPOTHESES. If a straight line from the vertex of a triangle bisects its base. Idem. If a straight line from the vertex of an isosceles triangle bisects its base. If a straight line bisects the angle opposite to the base of an isosceles triangle. If a straight line be drawn from the vertex of an isosceles triangle to any point in the base or the base produced. CONSEQUENCES. It also bisects the triangle. The sum of the squares on the two sides is equal in area to double the sum of the squares on half the base and on the bisecting line. It is perpendicular to it, And it bisects the opposite angle. It bisects the base, And it is perpendicular to it. The rectangle under the segments of the base is equal in area to the difference between the square on this line and the square on either side of the triangle. CONSEQUENCES. The difference of the squares on the sides is equal in area to the difference of the squares on the segments of the base. The sum of the squares on one side and the alternate segment is equal in area to the sum of the squares on the other side and the alternate segment. The difference of the squares on the sides is equal in area to twice the rectangle under the base and the distance of its middle point from the perpendicular. It will bisect the base, And also the angle opposite to the base. Double the rectangle under that side and the segment between the perpendicular and the base is equal in area to the square on the base. The square on the side subtending the obtuse angle is greater than the sum of the squares on the two sides which contain the obtuse angle, by double the rectangle under the side, which is produced, and the external segment between the obtuse angle and the perpendicular. The rectangle under one of those sides and the pro-duced part between the obtuse angle and the perpendicular, is equal in area to the rectangle under the other side and its produced part. The square on the side subtending that acute angle is less than the sum of the squares on the sides which contain that angle, by double the rectangle under the side to which the perpendicular is drawn, and the segment between the perpendicular and the acute angle. |