EXAMINATION PAPERS FOR REVISION. Note 1.–These papers are suitable for general purposes ; but have been constructed with a view to being worked during a revision of Euclid's text, as given in the larger print. Section I. here might then come after Section I. of the text; and similarly for the other sections. NOTE 2.—The student may be forewarned that a few of the following exercises consist of propositions of Euclid enunciated in different language to Euclid's own. A larger number consist of portions of Euclid's propositions ; and in each of these cases the proof should be given only so far as required by the portion which is set. SECTION I. PROPOSITIONS 1-6. I. 1. With a given straight line as one side, construct a rhombus, which shall be composed of two equilateral triangles. 2. X, Y, Z are three points in a straight line; from Z draw a straight line equal to XY, using the construction of Prop. II. 3. Given AB and C two straight lines, of which AB is the greater ; to cut off from BA a part equal to C, taking it from the B end of the line, after the manner of Prop. III. 4. If two triangles have two sides of the one and the angle which they contain respectively equal to two sides of the other and the angle which these contain, then the triangles shall be equal in area. 5. If two sides of a triangle are equal to one another, and these sides be produced, the two exterior angles of the triangle thus formed are equal to one another. (The exterior angles are those outside the triangle, “on the other side of the base.”) 6. ABC is a triangle on a level plane, having the angles B and C equal; we require the distance from C to A, but cannot measure it; while we can measure either of the other sides of the triangle. What shall we do? II. 1. XYZ is a straight line ; from X draw a straight line equal to Y2, using the construction of Prop. II. 2. Given AB and C two straight lines ; lengthen AB by a Euclidian method so as to make it equal to AB and C taken together. 3. If two sides and the included angle of one triangle be equal to two sides and the included angle of another, prove the third sides of the triangles are equal. 4. If C be the middle point of AB, and CD be at right angles to AB, prove that DAB is isosceles. 5. If FG, in the figure to Proposition V., be joined, then the angles AFG and AGF shall be equal. 6. With the same construction, show that the angles CFG and BGF will be equal. III. 1. XYZ is a straight line; from Y draw a straight line equal to XZ, using the construction of Prop. II. 2. Given a straight line AB, make an isosceles triangle upon it with each of the equal sides twice as long as AB. 3. If two triangles have two sides and the contained angle of the one equal to two sides and the contained angle of the other, each to each, they shall have their remaining angles equal in pairs; each of these pairs of angles having a pair of equal sides opposite to it. 4. In an isosceles triangle, show that the right line which bisects the vertical angle bisects the base, and is perpendicular to the base. 5. In an isosceles triangle the line joining the vertex to the middle point of the base is perpendicular to the base, and bisects the angle at the vertex. 6. If the angle ABC of the triangle ABC be double of the angle C, and BD bisect the angle ABC, meeting AC in D), show that DBC is isosceles. IV. 50°; 1. P is a point and QR a straight line ; from P draw a straight line twice as long as QR. 2. Two right-angled triangles are equal, if they have the sides containing the right angle in one respectively equal to the sides containing the angle in the other. 3. The angles A, B, C of a triangle are respectively 50°, 80°, which sides are equal ? 4. If C be the middle point of AB, the base of the triangle DAB, and DC be perpendicular to AB, then the angle ADB is bisected by DC. 5. Every quadrilateral whose diagonals bisect each other at right angles is a rhombus. 6. If the angles ABC, ACB at the base of an isosceles triangle be bisected by two straight lines BD, CD; then DBC will be isosceles. V. 1. ABC is an equilateral triangle, and D, E, F are points in AB, BC, CA respectively, such that AD, BE, CF are all equal ; if D, E, F be joined, the triangle thus formed will be equilateral. 2. Show that, if two straight lines bisect one another at right angles, any point in one of them is equidistant from the extremities of the other. 3. Prove that the straight lines which bisect two sides of a triangle at right angles meet in a point which is equidistant from the three angles. 4. In the figure to Euclid I. 5, if BG, CF intersect in H, and AH be joined, show that it bisects the angle A. (First show HB, HC equal by I. 6; then apply I. 4 to the triangles ABH, ACH.) 5. Deduce a construction for bisecting a given rectilineal angle, which requires no proposition of Euclid beyond I. 3; and whose proof requires no proposition beyond I. 6. 6. ABCD is a quadrilateral, having its four angular points on the circumference of a circle ; show that the sum of one pair of opposite angles is equal to the sum of the other pair. (Join A, B, C, D to the centre, and use 1. 5 four times.) SECTION II. PROPOSITIONS 7–12. VI. 1. If upon the same base and on the same side of it there be two triangles, and one of them be equilateral, prove that the other is not equilateral. 2. In equal circles, equal chords subtend equal angles at the centres. 3. In what particular case will the quadrilateral in the figure of I. 9 be a rhombus ? 4. Show how to find a point in side AB of any triangle ABC which is equidistant from the angles B and C. 5. What postulate should be applied in order to make the construction of I. 11 available when the given point is an extremity of the given line ? 6. If a straight line be drawn from the vertex of an equilateral triangle to the middle point of the base, prove that every point in it is equidistant from each base angle. (Apply I. 8 and I. 4 successively.) VII. 1. Two triangles are on opposite sides of a common base, and their other sides are respectively equal, and also one pair of equal sides are in one straight line; prove that each triangle is right-angled. (Use I. 8 and Def. 10.) 2. If two quadrilaterals have the four sides of the one respectively equal to those of the other in order, and the angle contained by two sides of one equal to the angle contained by the two sides equal to them of the other, then all the remaining angles of the figure shall be equal in pairs. (First prove two diagonals equal by I. 4, then use I. 8.) 3. Construct a right-angled triangle, given one side and the hypotenuse. 4. A is a given point, and BC a given straight line. Let the straight line which bisects AB at right angles meet BC in D. Show that D is equally distant from A and B. 5. A and B are given points above a given straight line CD; if there be a point in CD equally distant from A, B, show how to find it. 6. If two circles cut each other, the straight line joining the two points of intersection is at right angles to the line which joins their centres. VIII. 1. Given a finite right line of any length ; bisect it, and at the point of bisection erect a perpendicular to it. 2. Having given two points, find two others which shall be at a given distance from each of the first two. When will the problem become impossible ? 3. AB is a given straight line, and C, D are given points on opposite sides of AB; if there be a point in AB equally distant from C, D, find it. 4. If a given point be above a given straight line, but towards one end of it, and we do not wish to produce the line, prove that the following construction will give a perpendicular from the point to the line : Take any two points along the line; with each as centre, at the distance in each case of the given point, draw two arcs, meeting twice; join these two intersections of the arcs. 5. If BDC be drawn perpendicular to DA, and DC be made equal to BD; then two straight lines drawn from B and C to any one point in DA will be equally inclined to DA. 6. Through two given points on opposite sides of a given straight line draw two straight lines which shall meet in that given straight line, and include an angle bisected by it. SECTION III.-PROPOSITIONS XIII.-XVII. · IX. 1. If two straight lines meet at a point, and form there an angle equal to three-fourths of a right angle, what would be the magnitude of the adjacent angle which would be formed by producing ough the point either of the straight lines ? 2. Four straight lines are drawn from one point, and three successive angles thus formed are respectively two-thirds, seven-sixths, and fivesixths of a right angle in magnitude; show that two of the straight lines coincide, and that the other two do not. State the magnitude of the fourth angle. 3. The straight lines which bisect opposite vertical angles are in one and the same straight line. 4. Prove that any two exterior angles of a triangle are together greater than two right angles. 5. Show that only one perpendicular can be drawn to a given straight line from a given point without it. 6. Prove Euclid's Proposition XVII. without producing any side, but using a straight line joining any vertex to a point in the opposite side. X. 1. If four straight lines meet at a point, so as to make the opposite angles at that point equal, these straight lines are two and two in the same straight line. 2. If CD make with AB the adjacent angles ACD, DCB, prove that straight lines drawn bisecting these angles will contain a right angle. 3. From a given point let a straight line be drawn to a given straight line, making two angles with it; then, if one of these be acute, prove that the other will be obtuse. If from the point just mentioned a perpendicular be also drawn to the same given straight line, prove that it will subtend that angle of the two which is acute. 4. The supplement of any angle of a triangle is greater than either of the other two. 5. From the same point outside a given straight line it is not possible to draw to it three straight lines which are all equal. 6. Two finite right lines, of any lengths, being supposed radia in any directions from a common point; show that the angle they determine is equal to that determined by their two productions through the point. SECTION IV.-PROPOSITIONS XVIII.-XXV. XI. 1. In an obtuse-angled triangle, show that the longest side is opposite the obtuse angle. 2. If the base is the least side of an isosceles triangle, prove that the triangle is acute-angled. 3. Any three sides of a quadrilateral are greater than the fourth side. |