7. The vertical angle of an isosceles triangle is 40°; prove that the line bisecting its adjacent angle is parallel to the base.

8. How great is the angle between the lines bisecting two adjacent angles?

9. In a triangle the two angles are 40° and 80°; bisect the third angle, and from its vertex draw a perpendicular to the opposite side; how great is the angle between this and the bisecting line?

10. In a right-angled triangle one af the acute angles is ", how great is the other and how great are the angles in which the right angle is divided by a perpendicular from it to the hypothenuse?

11. Prove that the perpendicular from one end of the base of an isosceles triangle to the opposite side, forms an angle with the base, half as great as the vertical angle. 12. In a right-angled triangle the one angle is 60°; this is bisected, and from the vertex of the right angle a perpendicular is drawn to the hypothenuse. Prove that one of the triangles, thus formed, is equilateral.

13. The legs of one angle are perpendicular to the legs of another angle; prove that the two angles are equal or supplementary.

14. One of the angles at the base of an isosceles triangle, whose vertical angle is 36°, is bisected; prove that the two small triangles will be isosceles, and find the lengths of all the lines in the figure, when the sides of the given triangle are r and the base t.

15. In a triangle one angle is v; how great is the angle between the lines which bisect the two other angles? 16. Prove that an angle of a triangle is a right angle, when a line from its vertex to the middle point of the opposite side is half as great as this.

17. Prove that, when an angle of a right-angled triangle is 30°, then the lesser of the sides containing it is half as great as the hypothenuse.

18. The side AB of a triangle ABC is produced to D, SO that BD = BC. Prove that the line bisecting the angle B is parallel to the one joining DC.

19. Prove that in a right-angled isosceles triangle, the perpendicular from the vertex of the right angle to the hypothenuse is half as great as this.

20. The exterior angles of a triangle are bisected, thereby three triangles are formed, each having a side in common with the given triangle. Prove that the three triangles contain the same angles.

21. On one leg of any angle ABC, mark off any part AB, thereupon draw a line AD BC, and make AD AB. Prove that the line BD bisects the given angle. 22. In a triangle ABC the lines bisecting the angles A and B intersect in 0. Through 0 is drawn DE÷AB. Prove that DE AD + BE.

23. The demonstration of the proposition in 31 cannot be applied, if the two lines ry and zu cannot by prolongation be made to cut each other. How may it be demonstrated in this case?

24. Two angles have parallel legs; shew that the lines bisecting them are parallel or perpendicular to each other. 25. How many diagonals can be drawn in a polygon with n(n—3)

n sides? (

Ans.

2

26. In a quadrilateral the opposite sides of which are parallel the one angle is v; how great are the others? How great must the angle v be, in order that a circle can be described about the quadrilateral, and where will the centre of this circle lie?

27. Prove that the chord of an arc of 60° equals the radius. 28. In a circle two chords issue from the same point, and

cut off arcs of 120° and 80°. How great is the angle between the two chords?

29. In a circle a diameter AB is drawn, and a chord CD equal to the radius. How great is the angle between AC and BD, and between AD and BC?

30. In a triangle ABC, BC <BA.

=

With B as centre and

BC as radius a circle is described, cutting CA in E, BA in D. Shew that DEA B. 31. In a triangle ACB, C =

R.

With centre C describe a circle through the middle point of AB, cutting the hypothenuse or its prolongation the second time in D. Shew that the acute angle which CD makes with the hypothenuse is double of one of the angles of the triangle.

32. A circle has its centre on the circumference of another circle; the two circles cut one another in A and B; prove that the one arc AB contains half as many degrees as one of the other arcs AB.

33. In a circle two equal angles at the circumference BAC and DEF are drawn; shew that BF CD. 34. A triangle ABC is inscribed in a circle with centre 0; D is the middle point of the arc BC. Shew that the angle ADO is half as great as the difference between B and C.

35. Two chords EA and EB in a circle are produced to C and D, so that CD is parallel to the tangent at E. Prove that the opposite angles of ABCD are supplementary. 36. A is one of the points of intersection of two circles, of which the one passes through a point B, the other through a point C. Through A a line is drawn, cutting the first circle in D, the other in E. Prove that the angle between the lines BD and CE is constant (the same, in whatever way the line through A is drawn). 37. A circle touches one side of a triangle and the two other sides produced. Prove that the distances from the points of contact with the latter to their point of intersection equals half the perimeter of the triangle. 38. A triangle ABC, the angles of whieh are known, is inscribed in a circle. How great is the angle which the tangent touching the circle at A makes with BC? 39. Three small triangles are cut off from a triangle by tangents to its inscribed circle. Shew that the perimeters

of the three triangles are together equal to the perimeter of the given triangle.

40. Prove that the four lines bisecting the angles of any quadrilateral bound a quadrilateral, the opposite angles of which are supplementary.

41. Two circles cut each other in A and B. From A the diameters AC and AD are drawn; shew that C, B, and D lie in a straight line.

42. An angle has its vertex outside the circumference; one of its legs passes through the centre, and the part outside the circle of the other one equals the radius. Prove that the greater of the intercepted arcs is three times as great as the lesser.

43. Through each of the points of intersection of two circles

a straight line is drawn. Prove that the chords, joining the other points of intersection of these with the circles, are parallel.

44. Two circles with equal radii cut each other; shew that they divide each other into arcs which are respectively equal.

45. With one of the points of intersection of two equal circles as centre describe a circle cutting the given circles. Shew that two and two of the four points of intersection are in a straight line with the other point of intersection of the given circles.

46. Two equal circles cut each other in A and B. Through

A a line is drawn cutting the circles in D and E. Shew that DE is bisected by a circle on AB as diameter. 47. Between the circumferences of two circles, two lines are drawn through one of the points of intersection of the circles. Prove that the chords joining the extremities of the lines make the same angle in whatever way the two lines are drawn.

IV. RELATIONS BETWEEN THE LENGTH OF
STRAIGHT LINES.

36. The length of a straight line is measured, by stating how many times it contains another straight line, the unit, the length of which is supposed known; the foot it used as unit; it is divided into 12 inches, the inch into 12 lines (Duodecimal measure) or the inch into 10 parts (Decimal measure). 5 feet 7 inches 3 lines is written 5' 7" 3"".

A4

37. The greater side of a triangle has the greater angle

38. A greater angle of a triangle is subtended by a greater side.

When A > Z C, then BC> BA; for if not, then either BC BA or BC <BA, but from the first case it would follow that AC, and from the second that A < LO, ▲A< ZC, and both are in opposition to what was given.

A

C

This result may also be written AB AC-BC, so that in any

triangle one side is greater than the difference of the other two.