16. Define acute angle, figure, semicircle, parallelogram, and write out the axiom on which Euclid bases his reasonings on parallel lines, and also the axiom which is the definition of geometrical equality.

17. What figure is that of which the diagonals are equal, but do not intersect at right angles?

Propositions I.-XV. (inclusive).

1. Describe an equilateral triangle on a given finite straight line.

Same proposition. Is the demonstration direct or indirect? 2. Define a point and a straight line. From a given point draw a straight line equal to a given straight line.

Same proposition. What axioms are applied in this proposition? Same proposition. Draw the figure when the equilateral triangle is described on the side of the line opposite to that taken in your construction.

Same proposition. If you are able, prove this proposition by joining the point to the extremity of the given line which is farther removed from it, instead of the nearer one, as in Euclid.

3. Show how from the greater of two given straight lines to cut off a part equal to the less.

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 each other, 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.

Same proposition. If a triangle ABC be turned over about its side AB, show that the line joining the two positions of C is perpendicular to AB.

5. What is a plane triangle, and how many different kinds of triangles does Euclid name? Prove that the angles at the base of an isosceles triangle are equal, and if the equal sides be produced the angles on the other side of the base shall be equal.

Show that every equilateral triangle is equiangular.

6. If two angles of a triangle be equal to each other, the sides also which subtend, or are opposite to, the equal angles shall be equal to one another. What name is given to a

triangle which has two sides equal?

Same proposition. What property of equiangular triangles can be derived from this proposition? Give the steps necessary to prove this property.

Same proposition. A line which bisects the vertical angle of a triangle also bisects the base. Prove that the triangle is

isosceles.

Same proposition. What proposition is the converse of this.

Show that every equiangular triangle is equilateral.

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

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.

Is the equality of the triangles in all respects proved in this proposition? If not, how would you prove it?

Same proposition. If two circles cut each other, the lines joining their points of intersection is perpendicular to the line joining their centres.

Same proposition. From every point of a given line, the lines drawn to each of two given points on opposite sides of the lines are equal: prove that the line joining the given points will be bisected by the given line at right angles.

Same proposition. What is the hypothesis of this proposition?

9. Give Euclid's definition of a plane rectilineal angle. Show how to bisect a given rectilineal angle, that is, to divide it into two equal angles.

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

11. To draw a straight line at right angles to a given straight

line from a given point in the same. monstrate the corollary.

Also state and de

Demonstrate that two straight lines cannot have a common segment.

12. Show how to draw a straight line perpendicular to a given straight line of an unlimited length, from a given point without it. Explain why it is necessary to stipulate that the given line must be of unlimited length.

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

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.

Same proposition. If the angles are CBA, ABD, and BE bisects ABC, and BF bisects ABD, then FBE is a right angle.

Same proposition. Show that the vertex of an isosceles triangle, and the intersections of the bisections of the interior and exterior angles at the base are in the same straight line.

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

15. If two straight lines cut one another, the vertical, or opposite angles shall be equal. Deduce from this, that all the angles made by any number of straight lines meeting in one point are together equal to four right angles.

Same proposition. If the lines are AEB, CED, and FE, GE bisect the angles AEC, DEB; then FEG is a straight line. Same proposition. If AB, CD bisect each other in E, show that the triangles AED, BEC are equal in all respects. Same proposition. Give corollaries.

Propositions XVI.-XXVI. (inclusive).

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

Same proposition. Show that only one perpendicular can be drawn from a given point to a given straight line.

Same proposition. Prove this for both the interior angles.

Same proposition. Every straight line, drawn from the vertex of a triangle to the base, is less than the greater of the two sides.

Same proposition. Let ABC be the triangle in the above proposition. Bisect the angle ABC, produce the side AB to D, and bisect the angle CBD. Show that the two bisecting lines are at right angles.

17. Prove that any two angles of a triangle are together less than two right angles.

18. Give Euclid's definition of a plane rectilineal angle and of a triangle. Prove that the greater side of every triangle is opposite to the greater angle.

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

Same proposition. Hence show that the perpendicular is the shortest straight line that can be drawn from a given point to a given straight line.

20. Define a triangle, and mention the different kinds of triangles. Prove that any two sides of a triangle are together greater than the third side.

Same proposition. Any three sides of a quadrilateral are together greater than the fourth side.

Same proposition. How many triangles having two sides 5 feet and 6 feet long can be formed so that the third side shall contain a whole number of feet?

21. Prove that 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.

Same proposition. Is this a problem or theorem? Why?

22. Show how to make a triangle of which the sides shall be equal to three given straight lines, any two of which are together greater than the third. Explain why it is necessary that any two of the given straight lines should be greater than the third.

Define a triangle, and name the different classes of triangles. Make a triangle of which the sides shall be equal to three given straight lines. Can this be done with any three lines?

23. Show how at a given point in a given straight line to make a rectilineal angle equal to a given rectilineal angle.

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 those of the other; the base of that which has the greater angle shall be greater than the base of the other.

Same proposition. Explain clearly the reason for the restriction, 'Of the two sides DE, DF, let DE be not greater than DF.'

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

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 the equal angles in each; then shall the other sides be equal, each to each, and also the third angle of the one equal to the third angle of the other.

Propositions XXVII.-XLI. (inclusive).

27. When are straight lines said to be parallel to one another? Prove that if a straight line falling on two other straight lines make the alternate angles equal to each other, these two straight lines must be parallel.

Same proposition. What axiom does Euclid for the first time assume in proving the converse of this proposition?

Same proposition. If a straight line terminated by the sides. of a triangle be bisected, no other line terminated by the same two sides can be bisected in the same point.

Same proposition. A straight line is drawn bisecting an angle of a parallelogram: prove that there are formed by it and the sides of the parallelogram (or those produced) three triangles which are isosceles.

28. Prove that if a straight line falling upon two other straight lines makes the exterior angle equal to the interior and

b