11. State two methods of proving one thing twice another.

12. Define Polygon; Regular polygon.

13. State three principles of angle-sums in triangles or polygons. 14. State six methods of proving lines perpendicular.

15. State the basic axiom of all inequality.

16. State the axiom of inequality for straight lines. 17. State four methods of proving angles unequal. 18. State seven methods of proving lines unequal. 19. What are the five fundamental locus figures? 20. Define Locus.

21. State a method of finding the locus of points under given conditions.

22. State a method of finding a point under two given conditions. 23. State a method of proving a line is the locus of points under given conditions.

24. State four steps in an Analysis of Theorems.

25. What class of lines do not meet?

27. In Chapter V, the two main steps in most of the proofs are what?

28. If two straight lines are cut by a transversal making the alternate-interior angles equal, . . . what?

30. What are the two commonest methods of proving lines parallel? 31. Parallel lines cutting off equal parts on one transversal,

what?

32. What is the fundamental principle of angle-sums?

33. What is the commonest theorem of angle-sums?

34. What is the fundamental principle of perpendicular lines?

35. What is the commonest theorem?

36. The theorems of unequal angles are usually proved from what axiom?

37. The theorems of unequal lines are often proved from what axiom?

38. An exterior angle of a triangle is greater than 39. If two angles of a triangle are unequal,

40. State six methods of proving lines equal.

41. State six new methods of proving angles equal.

what?

what?

42. State a method of proving that two straight lines lie in one straight line.

43. What is the method of drawing an angle equal to a given angle?

44. The statement of a theorem is usually in what form?

45. If a theorem is true, is its converse necessarily true? Its opposite?

46. If a theorem is true, is the converse of the opposite true?

47. If a line is to be proved perpendicular to a given line, what equality may be proved?

48. In a proof, a statement is often obtained almost like the one desired; but one term may need to be changed. How is it possible to make this change in an equation or inequality?

49. In an inequality, give another method of making this change. 50. If two lines are in the same triangle, and are to be proved unequal, what two methods are probable?

51. In different triangles, what one method?

52. How is an exterior angle of a triangle formed?

53. How many can there be?

MISCELLANEOUS EXERCISES

1. If at the ends of a straight line perpendiculars to it are drawn, these perpendiculars cut off equal parts upon any oblique line drawn through the middle point of the given line.

2. If two triangles are congruent, the bisector of an angle of the one equals the bisector of the corresponding angle of the other.

3. If equal distances are measured on the sides of an angle from its vertex, the points thus obtained are equally distant from any point in the bisector of the angle.

4. If on the arms of an isosceles triangle equal distances are measured from the vertex, the lines joining these points to the opposite ends of the base are equal.

5. If the arms of an isosceles triangle are extended the same length through the vertex, the lines joining their ends to the ends of the base are equal.

6. If on the arms of an isosceles triangle equal distances are measured from the ends of the base, these points are equally distant from the opposite ends of the base.

7. If on the base of an isosceles triangle equal distances are measured from the ends of the base, the lines joining these points to the vertex are equal.

8. If the base of an isosceles triangle is extended both ways the same distance, the lines joining the ends to the vertex are equal.

9. The angle formed by two lines drawn within a triangle from the ends of one side is greater than the opposite angle.

10. The sum of the altitudes of a triangle is less than the perimeter. 11. In triangle ABC, AB is greater than AC, and the bisectors of angles B and C meet at O. Prove BO > CO. 12. In a quadrilateral ABCD, AB diagonal BD greater than diagonal AC.

=

CD, and < C > < B; prove

13. The line through the intersection of the bisectors of the base angles of a triangle and parallel to the base cuts off, on the sides, parts whose sum equals the parallel line.

14. If from a point outside an angle or its vertical angle perpendiculars are drawn to the sides of the angle, an angle is formed which is equal to the given angle.

15. The perpendicular from the vertex of the right angle of a right triangle to the hypotenuse divides the right angle into two parts equal respectively to the other angles of the triangle.

16. If a perpendicular is drawn from the middle point of the base of an isosceles triangle to one of the arms, this line makes with the base an angle equal to one-half the vertex angle.

17. Find each angle of a triangle if the first angle equals twice the second and the third equals three times the first.

18. If one angle of a triangle is xo, another angle yo, what expression represents the third angle?

19. If the vertex angle of an isosceles triangle is x°, what expression represents either base angle?

20. Trisect a right angle.

21. A line through the intersection of the diagonals of a parallelogram and ending in the sides is bisected by the intersecting point.

22. If each half of each diagonal of a parallelogram is bisected, the lines joining these points form parallelogram.

23. If the upper base of a trapezoid is one-half the lower base, the upper part of each diagonal is one-half the lower part.

24. The angle formed by the bisectors of the base angles of an isosceles triangle is equal to the exterior angle at one end of the base. 25. The diagonals of a rhombus divide the figure into four congruent triangles.

26. If a line is drawn across an angle perpendicular to the bisector of the angle, two congruent triangles are formed.

27. If perpendiculars are drawn from the ends of the base of a parallelogram, and the upper base is extended, two congruent right triangles are formed.

28. If two adjacent sides of a quadrilateral are equal and the diagonal bisects their included angle, the other two sides are equal.

29. In an isosceles triangle the vertex angle is 50°. Find the base angles.

30. How many degrees are there in each angle of an equiangular 9-gon? In each exterior angle?

31. If one acute angle of a right triangle is double the other, how many degrees in each?

32. If one acute angle of a right triangle is four times the other, how many degrees in each?

33. If the vertex angle of an isosceles triangle equals twice the sum of the base angles, how many degrees in each angle?

34. If one angle of a parallelogram is 40°, find each angle.

35. The exterior angle at the base of an isosceles triangle equals one-half the vertex angle plus a right angle.

36. If from any point in the base of an isosceles triangle, perpendiculars to the equal sides are drawn, they make equal angles with the base. 37. The bisectors of the angles of a trapezoid form a quadrilateral, two of whose angles are right angles.

38. The bisectors of the angles of a rectangle form a square.

39. If from any point in the base of an isosceles triangle parallels to the equal sides are drawn, the perimeter of the parallelogram equals the sum of the equal sides of the triangle.

40. The lines joining in order the middle points of the sides of a rectangle form a rhombus.

41. The bisectors of two exterior angles of a triangle and of the third interior angle meet in a point.

42. If AD is the bisector of ▲ A of ▲ ABC, and BE is || to AD and meets AC extended at E, prove ABE an isosceles A.

43. If AD is the bisector of exterior ▲ FAB of ▲ ABC, and BE is to AD and meets AC at E, prove ▲ ABE isosceles.

44. If two adjacent angles of a quadrilateral are right angles, the bisectors of the other angles are perpendicular.

45. If two opposite angles of a quadrilateral are right angles, the bisectors of the other angles are parallel.

46. The line joining the middle points of the diagonals of a trapezoid equals one-half the difference of the bases.

47. If two triangles are congruent, the corresponding medians are equal.

48. If two triangles are congruent, the corresponding altitudes are equal.

49. If one diagonal of a quadrilateral divides it into isosceles triangles, the other diagonal bisects two of the angles and is the perpendicular bisector of the first diagonal.

50. The bisectors of two exterior angles of a triangle form an angle equal to one-half the sum of the adjacent interior angles.

51. How many sides has an equiangular polygon, if an exterior angle equals 36°?

52. How many degrees in an interior angle of an equiangular polygon of 16 sides?

53. Two rectangles are congruent if they have two adjacent sides respectively equal.

54. If two isosceles triangles have the same base, the line joining their vertices bisects the angles.

55. If a quadrilateral has two opposite pairs of adjacent sides equal, the angle-bisectors meet in a point equally distant from the sides.