24. In a trapezoid, the straight line joining the middle points of the non-parallel sides is parallel to the bases, and is equal to one-half their sum. B H C Suggestion. Draw HG parallel to AB, and extend AD. DGF=CHF(Proposition VII.), and EFHB is a parallelogram, by Proposition XXX. B 25. If the sides of a trapezoid which are not parallel are equal, the base angles are equal and the diagonals are equal. 26. If through the four vertices of a quadrilateral lines are drawn parallel to the diagonals, they will form a parallelogram twice as large as the quadrilateral. 27. The three perpendiculars from the vertices of a triangle to the opposite sides meet in the same point. Suggestion. Draw through the three vertices lines parallel to the opposite sides of the triangle. By the aid of the three parallelograms ABCB', ABA’C, and ACBC', prove that the sides of A'B'C' are bisected by A, B, and C. See now Exercise 20. Α B F E 28. If a straight line drawn parallel to the base of a triangle bisects one of the sides, it also bisects the other side; and the portion of it intercepted between the two sides is equal to one-half the base. Suggestion. Draw DF parallel to AC. See now Proposition VII. and Proposition XXIX. D 29. The straight line joining the middle points of two sides of a triangle is parallel to the third side. (v. Exercise 28.) 30. The three straight lines joining the middle points of the sides of a triangle divide the triangle into four equal triangles. 31. In any right triangle, the straight line drawn from the vertex of the right angle to the middle of the hypotenuse is equal to one-half the hypotenuse. (v. Exercise 28.) D B E A 32. The straight lines joining the middle points of the adjacent sides of any quadrilateral form a parallelogram whose perimeter is equal to the sum of the diagonals of the quadrilateral (Exercise 29). 33. If E and Fare the middle points of the opposite sides, AD, BC, of a parallelogram ABCD, the straight lines BE, DF, trisect the diagonal AC (Exercise 28). 34. The four bisectors of the angles of a quadrilateral form a second quadrilateral, the opposite angles of which are supplementary. If the first quadrilateral is a parallelogram, the second is a rectangle. If the first is a rectangle, the second is a square. B D 35. The point of intersection of the diagonals of a parallelogram bisects every straight line drawn through it and terminated by the sides of the parallelogram. 36. If from each vertex of a parallelogram the same given distance is laid off on a side of the parallelogram, care being taken that no two distances are laid off on the same side, the points thus obtained will be the vertices of a new parallelogram. 37. If from two opposite vertices of a parallelogram equal distances are laid off on the sides adjacent to those vertices, the points thus obtained will be the vertices of a parallelogram. 38. The three medial lines of a triangle meet in the same point. Suggestion. Let O be the point of intersection of AD and BE, and H and G the middle points of OB and OA. Hence prove OD=}AD and OE =}BE. In like manner the point of intersection of AD and CF can be shown to cut off one-third of AD. D с Ꭰ B A A B D Ꭰ B Α Α' B F E D H 39. The intersection of the straight lines which join the middle points of opposite sides of any quadrilateral is the middle point of the straight line which joins the middle points of the diagonals (Exercise 29). ZA F B G SYLLABUS TO POSTULATES, AXIOMS, AND THEOREMS. POSTULATE I. Through any two given points one straight line, and only one, can be drawn. POSTULATE II. Through a given point one straight line, and only one, can be drawn having any given direction. AXIOM I. A straight line is the shortest line that can be drawn between two points. AXIOM II. Parallel lines have the same direction. PROPOSITION I. At a given point in a straight line one perpendicular to the line can be drawn, and but one. Corollary. Through the vertex of any given angle one straight linecan be drawn bisecting the angle, and but one. PROPOSITION II. All right angles are equal. PROPOSITION III. The two adjacent angles which one straight line makes with another are together equal to two right angles. Corollary I. The sum of all the angles having a common vertex, and formed on one side of a straight line, is two right angles. Corollary II. The sum of all the angles that can be formed about a point in a plane is four right angles. PROPOSITION IV. If the sum of two adjacent angles is two right angles, their exterior sides are in the same straight line. PROPOSITION V. If two straight lines intersect each other, the opposite (or vertical) angles are equal. PROPOSITION VI. Two triangles are equal when two sides and the included angle of the one are respectively equal to two sides and the included angle of the other. PROPOSITION VII. Two triangles are equal when a side and the two adjacent angles of the one are respectively equal to a side and the two adjacent angles of the other. PROPOSITION VIII. In an isosceles triangle the angles opposite the equal sides are equal. Corollary. The straight line bisecting the vertical angle of an isosceles triangle bisects the base, and is perpendicular to the base. PROPOSITION IX. Two triangles are equal when the three sides of the one are respectively equal to the three sides of the other. PROPOSITION X. Two right triangles are equal when they have the hypotenuse and a side of the one respectively equal to the hypotenuse and a side of the other. PROPOSITION XI. If two angles of a triangle are equal, the sides opposite to them are equal, and the triangle is isosceles. PROPOSITION XII. If two angles of a triangle are unequal, the side opposite the greater angle is greater than the side opposite the less angle. PROPOSITION XIII. If two sides of a triangle are unequal, the angle opposite the greater side is greater than the angle opposite the less side. PROPOSITION XIV. If two triangles have two sides of the one respectively equal to two sides of the other, and the included angles unequal, the triangle which has the greater included angle has the greater third side. |