opposite angle upon the same side of the line, or makes the interior angles upon the same side together equal to two right angles, the two straight lines shall be parallel to one another.

29. Prove that 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.

Same proposition. In the triangle ABC, AD bisecting the angle BAC meets BC in D, and DE, DF parallel to AC, AB respectively, meet AB, AC in E, F. Show that DE, DF are equal.

Same proposition. AB is parallel to CD, AD is bisected in E: show that any other straight line drawn through E to meet the two lines will be bisected in that point.

30. When are straight lines said to be parallel to each other? Prove that straight lines which are parallel to the same straight line are parallel to one another.

31. Draw a straight line through a given point parallel to a given straight line. From a given point draw a straight line such that the part of it included between two given parallel straight lines shall be of a given length. In what case would the construction fail?

Same proposition. Draw a line DE parallel to the base BC of a triangle ABC, so that DE is equal to the difference of BD and CE.

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

If the straight line bisecting the exterior angle be parallel to a side, show that the triangle is isosceles.

Same proposition. If the interior angle at one angular point of a triangle and the exterior angle at another be bisected by straight lines, the angle contained by the two bisecting lines is equal to half the third angle of the triangle.

State and prove the corollaries to the 32d proposition. Show by the proposition that if two straight lines be perpendicular to two other straight lines, each to each, the angle

included by the first pair is equal to the angle included by the second.

Same proposition. If the base of a triangle be produced both ways, the exterior angles are greater than two right angles by the vertical angle.

If the three sides of a triangle be produced, the sum of the exterior angles is equal to four right angles.

Same proposition. If the alternate sides of any polygon be produced to meet, the angles formed by these lines, together with eight right angles, are together equal to twice as many right angles as the figure has sides.

Same proposition. ABC is a triangle right-angled at A, and the angle B is double of the angle C. Show that the side CB is double of the side AB.

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. ABCD is a parallelogram whose diagonals AC, BD intersect in C: show that if the parallelograms AOBP, DOCQ be completed, the straight line joining P and Q passes through O.

34. Define a parallelogram and its diameter. Show that the opposite sides and angles of parallelograms are equal to one another, and that the diameter bisects them, that is, divides them into two equal parts.

Same proposition. Prove that any line terminated by two opposite sides of a parallelogram and passing through the intersection of the diagonals is bisected in that point.

Same proposition. The line drawn from the right angle of a triangle to the middle of the hypotenuse is equal to half the hypotenuse.

Same proposition. If the two diagonals are drawn, show that a parallelogram will be divided into four equal parts; in what case will the diagonal bisect the angle?

35. Parallelograms on the same base and between the same parallels are equal to one another. What is the converse of this proposition?

Same proposition. Describe a parallelogram equal to a given square having an angle equal to half a right angle.

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

37. Triangles upon the same base and between the same parallels are equal to one another. Any point P is taken in the line joining an angular point A of a triangle to the middle point of the opposite side BC: prove that the triangles APB and APC are equal.

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

The equal sides AB, AC of an isosceles triangle are produced to D, E, so that AD=2AB, AE= 2AC.

Show that

CD and BE are trisected at their point of intersection.

Same proposition. ABC, ABD are two equal triangles on the same base and on opposite sides of it. If CD meets AB in E, then CE, ED are also equal.

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

AB, CD are parallel straight lines; AD, BC meet in E, and CD is produced to F, so that the triangles CEF, ACD are equal. Show that FB is parallel to AD.

40. Equal triangles upon equal bases in the same straight line, and towards the same parts, are between the same parallels. Are the angles of these triangles equal, each to each?

41. Define a parallelogram. Prove that if a parallelogram and a triangle be upon the same base, and between the same parallels, the parallelogram shall be double of the triangle.

Same proposition. Hence deduce that two triangles formed by drawing straight lines from any point within a parallelogram to the extremities of its opposite sides, are together half of the parallelogram.

Propositions XLII.-XLVIII. (inclusive).

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.

Same proposition. If in the figure 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.

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

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.

45. Show how to describe a parallelogram equal to a given rectilineal figure, and having an angle equal to a given rectilineal angle.

46. Show how to describe a square upon a given straight line.

Describe a square upon a given straight line. Deduce clearly from the working of this problem that every parallelogram which has one right angle has all its angles right angles.

Same proposition. Also to describe a square on a given straight line as a diameter.

Same proposition. On a given straight line describe an isosceles triangle having each of the sides equal to a given straight line.

Same proposition. Through a given point on a side of a square, draw a line to the opposite side which shall bisect the square.

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.'

Same proposition. What is the length of a ladder which will just reach from the edge of a ditch 5 yards wide, to the top of a wall on the opposite side 12 yards high?

48. Prove that if the square described upon one of the sides of a triangle be equal to the sum of the squares described upon the other two sides of it, the angle contained by these two sides is a right angle.

Deductions.

1. The angles at the base of an isosceles triangle are equal. Show from this that the opposite angles of a rhombus are equal to each other.

2. Any two sides of a triangle are together greater than the

third side. Hence prove that the sides of a four-sided figure are together greater than the two diagonals.

3. Through a given point within an angle BAC to draw a straight line cutting off equal parts from AB, AC.

4. By the method of the first proposition describe on a given finite straight line an isosceles triangle the sides of which shall be each equal to twice the base.

5. Describe on a given straight line, as diagonal, a rhombus, each of whose sides is equal to that line, by the method employed in the first proposition.

6. If two right-angled triangles have two sides containing an acute angle of the one equal to two sides containing an acute angle of the other, show that the triangles are equal in all respects.

N.B.-This is a case of the equality of triangles not proved by Euclid. The proof is given in Potts' Euclid, in the note on I. 26, at the end of the First Book.

7. If a point P be taken inside a quadrilateral ABCD, prove that the sum of the distances of P from the angular points is the least possible when P is situated at the intersection of the diagonals.

8. The sum of the diagonals of a quadrilateral is less than the sum of any four lines that can be drawn from any point whatever (except the intersection of the diagonals) to the four angles.

9. Prove that the sum of the distances of any point from the angular points of a quadrilateral is greater than half the perimeter of the quadrilateral.

10. The diagonals of a rhombus bisect each other at right angles.

11. Describe a square which shall be equal to the difference between two given squares.

12. Find a point in the diagonal of a square produced, from which, if a straight line be drawn parallel to any side of the square, and meeting another side produced, it will form, together with the produced diagonal and produced side, a triangle equal to the square.