SECTION VIII. PROPOSITIONS XXXIII., XXXIV. XXIII. A quadrilateral figure will be a parallelogram1. If two opposite sides be both equal and parallel ; or 2. If its opposite sides are equal; or 3. If its opposite angles are equal; or • 4. If its diagonals bisect one another. 5. If a quadrilateral be a parallelogram, its diagonals bisect one another. 6. Through a given point to draw a straight line cutting two parallel straight lines so that the portion intercepted between them shall be equal to a given straight line. XXIV. 1. If each diameter of a quadrilateral bisect two opposite angles, it must be a rhombus. - 2. If the diagonals of a parallelogram are equal, all its angles are equal and right. 3. In a parallelogram, of any magnitude and form, show that each of the four angles, and also each of the four sides, is equal to its opposite. 4. By aid of I. 34 or otherwise, draw a chord of a given angle, which shall be at once parallel and equal to a given finite right line. 5. Given two straight lines which meet, to find a point such that the perpendiculars let fall from it upon the two given lines shall be respectively equal to two other given straight lines. 6. Construct a parallelogram which shall have two adjacent sides and the diagonal through their point of intersection equal to three given -straight lines of finite lengths. SECTION IX. PROPOSITIONS XXXV.-XLII. XXV. 1. A parallelogram being given which is not rectangular, construct a rectangle equal to it. 2. Show how to bisect a triangle by a straight line drawn through the vertex. 3. If a quadrilateral figure be bisected by each diagonal, it is a parallelogram. 4. A straight line drawn through the middle point of the diameter of a parallelogram, and terminated by two opposite sides, bisects the parallelogram. 5. The four triangles into which a parallelogram is divided by its diagonals are equal. 6. If, in the figure of I. 42, the triangle be equilateral, and the given angle two-thirds of a right angle, the perimeters of the parallelogram and the triangle are also equal. XXVI. 1. A parallelogram being given which is not equilateral, construct a rhombus equal to it in area. 2. ABCD is a quadrilateral, having BC parallel but not equal to AD; a parallelogram is formed by drawing through the middle of DC a straight line parallel to AB to meet the parallel sides, produced if necessary; show that this figure is equal to the original quadrilateral. 3. If two triangles have two sides of the one equal to two sides of the other, each to each, and the angles contained by those sides supplementary, the triangles are equal in area. 4. Bisect a parallelogram by a straight line drawn through any given point outside of it. 5. If from a point in one diagonal of a parallelogram straight lines be drawn to the opposite angles, these will form two pair s of equal triangles. . 6. Describe a parallelogram equal to a given triang le, and such that the sum of its sides shall be equal to the sum of the sides of the triangle. XXVII. 1. Make two parallelograms which shall have their sides equal, each to each, but the area of one double that of the other. 2. If ABCD be a quadrilateral whose diagonals meet in E, and the triangles AED, BEC are equal; show that AB and CD are parallel. 3. The straight line which joins the middle points of two sides of a triangle is parallel to the base and equal to half of it. 4. Straight lines which join the middle points of adjacent sides of a quadrilateral form a parallelogram. 5. If the middle points of the sides of a triangle be joined, the triangle is divided into four equal triangles. 6. If straight lines are drawn from any point within a parallelogram to the angles, the two triangles thus formed, upon either pair of opposite sides of the figure as bases, are together equal to half the whole figure. XXVIII. 1. Prove that a parallelogram is equal in area to a rectangle which has its base and altitude equal to the base and altitude of the parallelogram. 2. Two triangles, having a common base, being supposed to have their two vertices in a common parallel to the base ; show that the four parallelograms on the same base, having their four sides for diagonals, are equal in area. 3. By aid of the preceding, or otherwise, construct on a given base a triangle of given area, having its vertex on a given indefinite right line not parallel to the base; and determine the number of solutions. 4. If through the middle point of a side of a triangle a straight line be drawn parallel to the base, it will bisect the other side, and that part of it which is intercepted between the sides of the triangle will be equal to half the base. 5. Prove that a quadrilateral which has two opposite sides parallel is equal in area to a rectangle between the same parallels on a base equal to half the sum of the parallel sides of the given quadrilateral. 6. On a given finite right line of any length as base, construct a rectangle which shall be equal to a given square. SECTION X. PROPOSITIONS XLIII.-XLV. XXIX. 1. The parallelograms about the diagonal of a square are likewise squares. 2. To one of the sides of an equilateral triangle apply an equal parallelogram having one of its angles equal to that of the triangle. 3. Construct a parallelogram equal to the difference of two rectilineal figures. 4. If through a point O within the parallelogram ABCD two straight lines are drawn parallel to the sides, and the parallelograms OB, OD are equal, the point 0 is in the diagonal AC. 5. To a given straight line apply a triangle equal to a given parallelogram, and having an angle equal to a given rectilineal angle. 6. Describe a triangle upon a given base equal in area to a given triangle. SECTION XI. PROPOSITIONS XLVI.-XLVIII. XXX. 1. Squares constructed upon equal straight lines are equal themselves, and conversely. 2. Describe a square equal to the double of a given square. 3. The square upon a given straight line is equal to four times the square upon its half. 4. If a perpendicular be drawn from the vertex upon the base of any triangle, the difference between the squares on the sections of the base is equal to the difference between the squares on the two sides of the triangle. 5. Construct a square equal to the difference between two given squares. 6. Prove that the square on the side subtending an acute angle of a triangle is less than the square upon the sides containing that angle. XXXI. 1. Prove that two right-angled triangles are equal in all respects if the hypotenuse and a side in the one are respectively equal to the hypotenuse and a side in the other. 2. Describe a square equal to three times a given square; also a square equal to the sum of three given squares. 3. The squares upon the sides of a rhombus are together equal to the squares upon its diagonals. 4. If from the middle point of one of the sides of a right-angled triangle a perpendicular be drawn to the hypotenuse, the difference of the squares on the segments so formed is equal to the square on the other side. 5. If a straight line be divided into any two parts, the square upon the whole line is greater than the sum of the squares on the parts. 6. The square upon the side subtending an obtuse angle of a triangle is greater than the sum of the squares upon the sides which contain the obtuse angle. SCIENCE AND ART DEPARTMENT'S PAPERS, Set in the Examinations in Pure Mathematics, Stage I. Note.—The numbering of the questions is given as in the actual papers ; in which six questions in Arithmetic come before these in Geometry, and six in Algebra come after. The numbers in brackets are those assigned by the examiner, and indicate the relative importance of the questions. 1875. 7. Prove that the angles at the base of an isosceles triangle are equal to one another. (8) 8. Prove that any two sides of a triangle are together greater than the third. (8) 9. Let A, B, C be three points in a straight line taken in order, and D a any other point not in that line. Join BD. Show that the lines which bisect the angles ABD and DBC are perpendicular to one another. (10) 10. Show that a line drawn from the angle (or vertex) of a triangle so as to bisect the base, also bisects the triangle. (12) 11. Prove that the diagonals of a parallelogram bisect one another. (14) 12. Show that, if a straight line meets two parallel straight lines, the alternate angles are equal. Write out at full length what definition of parallelism, and what axiom (if any), you use to prove this. No marks will be given for a mere proof if this be not done. (14) 1876. (8) 7. If two triangles have two sides of the one equal to two sides of the other, each to each, and have also the angles contained by those sides equal to one another, they shall also have their bases or third sides equal. (8). 8. Any two angles of a triangle are together less than two right angles. 9. The opposite sides and angles of a parallelogram are equal to one another. (9) 10. If two isosceles triangles have the same base, the line which joins their summits is at right angles to the base. (12) 11. ABC is an equilateral triangle. On AB take any distance AF less than AB. On BC take BD equal to AF, and on CA take CE equal to AF. Join AD, BE, and CF. Show that these three lines either meet in one point, or enclose an equilateral triangle. (14) 12. What use does Euclid make in his First Book of the tenth axiom, namely, that two straight lines cannot enclose a space ? (14) 1877. 7. Prove that the three angles of a triangle, taken together, are equal to two right angles. (8) 8. Through a given point outside a straight line draw another straight line which shall make with the first an angle equal to a given angle. How many such lines can be drawn ? (9) 9. Prove that, if the squares upon two of the sides of a triangle, taken together, are equal to the square on the third side, the triangle is rightangled. Which side is the right angle opposite to ? (9) 10. Prove that, if the four sides of a parallelogram are all equal, the two diagonals bisect, and are perpendicular to, one another. (12) 11. The equal sides of an isosceles triangle are each 13 inches long, and the base is 10 inches. Find the length of the shortest line which can be drawn from the vertex to the base. (12) |