QUESTIONS AND EXERCISES

ON

BOOK I.

DEFINITIONS.

1. Define a point. II. Define a line. III. What are the extremities of a line? IV. What is a straight line? v. Define a superficies. VI. The extremities of a superficies. vII. A plane superficies. VIII. A plane angle. IX. A plane rectilineal angle. x. A right angle, and a perpendicular. XI. An obtuse angle. XII. An acute angle. XIII. What is a term, or boundary? XIV. Define a figure. xv. A circle. XVI. What is the centre of a circle? XVII. What is the diameter of a circle? XVIII. What is a semicircle? XIX. What is the centre of a semicircle? xx. Define a rectilineal figure. XXI. A trilateral figure or triangle. XXII. A quadrilateral figure. XXIII. A multilateral figure, or polygon. XXIV. An equilateral triangle. XXV. An isosceles triangle. XXVI. A scalene triangle. xxvII. A right-angled triangle. XXVIII. An obtuse-angled triangle. XXIX. An acute-angled triangle. xxx. A square. XXXI. An oblong. XXXII. A rhombus. XXXIII. A rhomboid. xxxiv. A trapezium. xxxv. What are parallel straight lines?

POSTULATES.

1. Enunciate the first postulate. II. Enunciate the second postulate. III. Enunciate the third postulate..

AXIOMS.

1. How are things which are equal to the same thing related to one another ?

II. If equals be added to equals, how are the wholes related to one another ?

III. If equals be taken from equals, how are the remainders related to one another?

IV. If equals be added to unequals, how are the wholes related to one another ?

v. If equals be taken from unequals, how are the remainders related to one another?

VI. How are the doubles of the same thing related to one another?

VII. How are the halves of the same thing related to one another?
VIII. When are magnitudes equals?

IX. How is the whole related to its part?

x. How are right angles related to one another?

XI. Under what circumstances will two straight lines meet, if they are continually produced, and on what side will they meet?

ENUNCIATIONS AND COROLLARIES OF THE PROPOSITIONS OF BOOK I.

FOR EXAMINATION APART FROM THE TEXT.

Prop. 1. To describe an equilateral triangle upon a given finite straight line.

Prop. 2. From a given point to draw a straight line equal to a given straight line.

Prop. 3. From the greater of two given straight lines to cut off a part equal to the less.

Prop. 4. If two triangles have two sides of the one equal to two sides of the other, each to each, and have likewise the angles contained by those sides equal to one another, they shall likewise have their bases, or third sides, equal; and the two triangles shall be equal; and their other angles shall be equal, each to each, viz., those to which the equal sides are opposite.

Prop. 5. The angles at the base of an isosceles triangle are equal to one another; and if the equal sides be produced, the angles upon the other side of the base shall be equal.

Cor. Every equilateral triangle is also equiangular.

Prop. 6. If two angles of a triangle be equal to one another, the sides also which subtend, or are opposite to the equal angles, are equal to one another. Cor. Every equiangular triangle is also equilateral.

Prop. 7. Upon the same base, and on the same side of it, there cannot be two triangles having their sides terminated in one extremity of the base, equal to one another, and likewise those which are terminated in the other extremity.

Prop. 8. If two triangles have two sides of the one equal to two sides of the other, each to each, and have likewise their bases equal; the angle which is contained by the two sides of the one shall be equal to the angle contained by the two sides equal to them of the other.

Prop. 9. To bisect a given rectilineal angle; that is, to divide it into two equal angles.

Prop. 10. To bisect a given finite straight line; that is, to divide it into two equal parts.

Prop. 11. To draw a straight line at right angles to a given straight line, from a given point in the same.

Prop. 12. To draw a straight line perpendicular to a given straight line of unlimited length, from a given point without it.

Prop. 13. The angles which one straight line makes with another upon one side of it, are either two right angles, or are together equal to two right angles.

Prop. 14. If, at a point in a straight line, two other straight lines, upon the opposite sides of it, make the adjacent angles together equal to two right angles, these two straight lines shall be in one and the same straight line.

Prop. 15. If two straight lines cut one another, the vertical or opposite angles shall be equal.

Cor. 1. If two straight lines cut one another, the angles they make at the point where they cut, are together equal to four right angles.

Cor. 2. All the angles made by any number of lines meeting in one point are together equal to four right angles.

Prop. 16. If one side of a triangle be produced, the exterior angle is greater than either of the interior opposite angles.

Prop. 17. Any two angles of a triangle are together less than two right angles.

Prop. 18. The greater side of every triangle is opposite to the greater angle. Prop. 19. The greater angle of every triangle is subtended by the greater side, or has the greater side opposite to it.

Prop. 20. Any two sides of a triangle are together greater than the third side.

Prop. 21. If from the ends of a side of a triangle, there be drawn two straight lines to a point within the triangle, these shall be less than the other two sides of the triangle, but shall contain a greater angle.

Prop. 22. To make a triangle of which the sides shall be equal to three given straight lines, but any two whatever of these must be greater than the third.

Prop. 23. At a given point in a given straight line, to make a rectilineal angle equal to a given rectilineal angle.

Prop. 24. If two triangles have two sides of the one equal to two sides of the other, each to each, but the angle contained by the two sides of one of them greater than the angle contained by the two sides equal to them of the other; the base of that which has the greater angle shall be greater than the base of the other.

Prop. 25. If two triangles have two sides of the one equal to two sides of the other, each to each, but the base of the one greater than the base of the other; the angle contained by the sides of that which has the greater base, shall be greater than the angle contained by the sides equal to them of the other.

Prop. 26. If two triangles have two angles of the one equal to two angles of the other, each to each; and one side equal to one side, viz., either the sides adjacent to the equal angles, or the sides opposite to equal angles in each; then the other sides shall be equal, each to each, and also the third angle of the one to the third angle of the other.

Prop. 27. If a straight line falling upon two other straight lines, make the alternate angles equal to one another; these two straight lines shall be parallel.

Prop. 28. If a straight line falling upon two other straight lines, make the exterior angle equal to the interior and opposite angle upon the same side of the line; or make the interior angles upon the same side together equal to two right angles; the two straight lines shall be parallel to one another.

Prop. 29. If a straight line fall upon two parallel straight lines, it makes the alternate angles equal to one another; and the exterior angle equal to the interior and opposite angle upon the same side; and likewise the two interior angles upon the same side together equal to two right angles.

Prop. 30. Straight lines which are parallel to the same straight line, are parallel to each other.

Prop. 31. To draw a straight line through a given point parallel to a given straight line.

equal

Prop. 32. If a side of any triangle be produced, the exterior angle to the two interior and opposite angles; and the three interior angles of every triangle are together equal to two right angles.

Cor. 1. All the interior angles of any rectilineal figure, together with four right angles, are equal to twice as many right angles as the figure has sides. Cor. 2. All the exterior angles of any rectilineal figure (made by producing the sides successively in the same direction), are together equal to four right angles. Prop. 33. The straight lines which join the extremities of two equal and parallel straight lines towards the same parts, are also themselves equal and parallel.

Prop. 34. The opposite sides and angles of a parallelogram are equal to one another, and the diameter bisects it, that is, divides it into two equal parts.

Prop. 35. Parallelograms upon the same base, and between the same parallels, are equal to one another.

Prop. 36. Parallelograms upon equal bases, and between the same parallels, are equal to one another.

Prop. 37. Triangles upon the same base, and between the same parallels, are equal to one another.

Prop. 38. Triangles upon equal bases, and between the same parallels, are equal to one another.

Prop. 39. Equal triangles upon the same base, and upon the same side of it, are between the same parallels.

Prop. 40. Equal triangles upon equal bases in the same straight line, and towards the same parts, are between the same parallels.

Prop. 41. If a parallelogram and a triangle be upon the same base, and between the same parallels; the parallelogram shall be double of the triangle. Prop. 42. To describe a parallelogram that shall be equal to a given triangle, and have one of its angles equal to a given rectilineal angle.

Prop. 43. The complements of the parallelograms, which are about the diameter of any parallelogram, are equal to one another.

Prop. 44. To a given straight line to apply a parallelogram, which shall be equal to a given triangle, and have one of its angles equal to a given rectilineal angle.

Prop. 45. To describe a parallelogram equal to a given rectilineal figure, and having an angle equal to a given rectilineal angle.

Cor. To a given straight line, to apply a parallelogram, which shall have an angle equal to a given rectilineal angle, and shall be equal to a given rectilineal figure.

Prop. 46. To describe a square upon a given straight line.

Cor. Every parallelogram that has one right angle, has all its angles right angles.

Prop. 47. In any right-angled triangle, the square which is described upon the side subtending the right angle, is equal to the squares described upon the sides which contain the right angle.

Prop. 48. If the square described upon one of the sides of a triangle, be equal to the squares described upon the other two sides of it; the angle contained by these two sides is a right angle.

GEOMETRICAL EXERCISES ON BOOK I.

The following Exercises, which have been selected on account of their simplicity, and their value as geometrical facts useful to the learner in more advanced investigations, have been arranged, up to Ex. 48, as far as possible in the order of the Propositions. After that number they are of a more miscellaneous character, but it is believed that none of them are beyond the capacity of the student who has gone carefully through the First Book.

1. In the figure of Euclid, Book I., Prop. I., describe an equilateral triangle upon the other side of AB.

2. If in the same figure, AB be produced both ways to meet the circles in D and E, and from C, the lines CD and CE be drawn; show that the triangle CDE is isosceles.

3. In the same figure, if the circles intersect in F on the other side of AB, and AF, BF be drawn ; prove that ACBF is a rhombus.

4. In the same figure, if the given line is produced to meet either of the circles in P; show that P and the points of intersection of the circles, are the angular points of an equilateral triangle.

5. If in the same figure, CA, CB be produced to meet the circumference in D, E, and F be the other point of intersection of the circles, show that DF, EF are in one line.

6. Upon a given straight line describe an isosceles triangle that shall have each of its sides double of the base.

7. Prove by superposition, that if two squares have one side of the one equal to one side of the other, the squares are equal in all respects.

8. In the figure of Prop. 1. 5, if FC and BG meet at H, show that AH bisects the angle BAC.

9. In the same figure, if FC and BG meet at H, show that FH and GH are equal.

10. On a given straight line to describe an isosceles triangle, of which the perpendicular height is equal to the base.

11. In an isosceles triangle, the straight line which bisects the vertical angle also bisects the base, and is perpendicular to it.

12. In an isosceles triangle, the line drawn from the vertex to the middle point of the base bisects the vertical angle, and is perpendicular to the base.

13. If, in a triangle, the perpendicular from the vertex on the base bisect the base, the triangle is isosceles.

14. The opposite angles of a rhombus are equal.