(c) If the members of an inequality be multiplied or be divided by a positive quantity, the inequality will subsist in the same sense; but if multiplied or divided by a negative quantity, the inequality will be reversed. 7. Show how to construct a triangle, having given two angles and the side opposite one of them without having found the third angle. 8. Show that the difference of two sides of a triangle is less than the third side. 9. Through a point to draw a line parallel to a given line, and show that only one such line can be drawn. 10. Show how we may, with ruler and compass, erect a perpendicular at any point of a line, and how we may let fall a perpendicular from a point. 11. Prove the converse of Ex. 2, § 18. NOTE. A proposition is a statement of a relation said to exist, and is in the general form of subject and predicate. D The converse of a proposition is also a proposition; but with relations of subject and predicate reversed. Exercise 2, § 18, as a formal proposition would be: (If the two lines AC and BD are parallel and are intersected by a third line HE) (then will) (the angles BAC and ABD be supplementary). The converse would be: (If two lines are met by a third line so as to make the interior angles on one side of the secant supplementary) (then will) (the two lines so situated with respect to the third be parallel). The Reductio ad Absurdum method is particularly well adapted to the determination of the truth or falsity of the converse of a proposition. CHAPTER III. 29. Definitions. A triangle is called: Equiangular, when the three angles are equal to each other. Equilateral, when the three sides are equal to each other. Isosceles, when two sides are equal to each other and not equal to the third side. Scalene, when there is not any equality between sides. АД FIG. 36. Special names given to parts of triangles are: Base, which may be any one side. Base angles, which are the angles adjacent to the base. Vertex, which is the point of intersection of the sides, not considered the base. Vertex angle, which is the angle of the triangle at the vertex. Hypothenuse, which is the side opposite a right angle. Altitude, which is the perpendicular from vertex to base. 30. THEOREM. If at the middle of a side of an equilateral triangle a perpendicular be erected, it will pass through the opposite vertex. If at M, the middle point of the side AB, a perpendicular be erected, it will contain all points that are equally distant from A and B. C is 4. Show that an equilateral triangle is equiangular. 5. Establish the converse. 31. THEOREM. If a perpendicular be erected at the middle point of the non-equal side of an isosceles triangle, it will pass through the opposite vertex. The proof is the same as in § 30. Exercises. - 1. Show that the angles opposite the equal sides are equal. 2. Show that the angle at the vertex is bisected by the perpendicular. 3. Show that if perpendiculars be let fall from the vertices of the equal angles, they will be equal to each other. Solution. Let ACB represent the isosceles triangle; and AP and BR the perpendiculars. If the perpendiculars are equal, the AAPC and BRC will be equal because they will then have a perpendicular and an hypothenuse of a right triangle the same in each. Does this relation (the equality of the triangles) exist without the consideration of the perpendicular? Hence by (§ 23, Ex.), ▲ APC = ^ BPC. AP and BR are corresponding parts and are equal. The question has thus been answered: The necessary relation does exist, and the problem is solved. Q. E. D. 4. Solve the same problem, using for the figure an acuteangled triangle. 5. Show that if two angles of a triangle are equal, the sides opposite those angles are equal, i.e. the triangle will be isosceles. NOTE. A problem is something proposed to be done. The solution is the finding of sufficient previously established relations to warrant the doing. |