Ex. 10. If a diagonal of a quadrilateral ABCD bisects two of its angles, it is perpendicular to the other diagonal and bisects it. Suggestions. 1. Let AC bisect LA and LC; prove AC1 BD and AC bisects BD. 2. Try to prove AB AD and BC= DC. Ex. 11. In the adjoining figure, if AO, BO, and CO are extended to Z, Y, and I respectively, so that AO OZ, BO=0Y, and CO = OX, then AABC=AZYX. = (First prove AB = ZY, BC = XY, and AC = XZ.) B A After proving ▲ ABC≈ ^ XYZ, what angle does / BCA equal? B Y Ex. 12. Prove that homologous medians of congruent triangles are equal. Suggestion. Read the suggestions for Ex. 6, p. 273. Ex. 13. If two triangles have two sides and the median to one of them equal respectively to two sides and the corresponding median of the other, the triangles are congruent. Suggestion. Read the note following § 77. Ex. 14. Construct the perpendicular-bisector of a segment taken along the lower edge of the paper. Ex. 15. Draw any angle and construct its bisector. Through its vertex construct a line perpendicular to the bisector. Prove that this last line makes equal angles with the sides of the given angle. Ex. 16. Construct a line through a given point within a given acute angle, which will form with the sides of the angle an isosceles triangle. Suggestion. If ▲ ABC represents the desired triangle, and AH bisects CAB, then CB 1 AH. Try now to work toward this figure if only FAG and point D within it are given. F B E A H D joining figure, prove AB CD and also AC || BD. Suggestion. Recall § 95. Ex. 21. In building stairs two or more stringers are required. To make a stringer having a 9-in. tread and a 6-in. riser, a carpenter uses his square as in the figure below. For each step he places his square so that the 9-in. mark and the 6-in. mark fall along the edge of the board from which he is cutting the stringer. Will the treads all be parallel? Why? Will the risers all be parallel? Why? Will each riser be perpendicular to its tread? Why? tread Ex. 22. One triangle used by draughtsmen has an angle of 90° and an angle of 60o. Why should it be called a "60-30" triangle ? Ex. 23. The other triangle used by draughtsmen has an angle of 90° and an angle of 45°. How large is the remaining angle ? Ex. 24. In a ▲ ABC, if ZA = 90°, and B = C, how large are LB and C ? Ex. 25. Find the three angles of a triangle if the second is four times the first, and the third is seven times the first. (Algebraic solution.) Ex. 26. Find the three angles of a triangle if the second exceeds the first by 40°, and the third exceeds the second by 40°. Ex. 27. The vertical angle of an isosceles triangle is n degrees. Express each of the base angles. Ex. 28. One base angle of an isosceles triangle is n degrees. Express each of the other angles of the triangle. Ex. 29. Determine by construction the angle C of a ▲ ABC if Z A and B are the angles given in Ex. 61, Book I. Ex. 30. Prove that two isosceles triangles are congruent when the vertical angle and the base of one are equal respectively to the vertical angle and the base of the other. Suggestion. - Prove the homologous base angles also are equal. Ex. 31. If one acute angle of a right triangle is 35°, how large is the other acute angle? Ex. 32. If perpendiculars be drawn from any point in the base of an isosceles triangle to the equal sides, they make equal angles with the base. Suggestion. The proof is based on § 37 and § 109. Ex. 33. Prove that the altitude drawn to the hypotenuse of a right triangle divides the right angle into two parts which are equal respectively to the acute angles of the right triangle. Ex. 34. If two opposite angles of a quadrilateral are equal and if the diagonal joining the other two angles bisects one of them, then it bisects the other also. Ex. 35. If two triangles have two angles and the bisector of one of these angles equal respectively to two angles and the corresponding bisector of the other, the triangles are congruent. Suggestion. Recall the note following § 77. Ex. 36. If two triangles have two sides and the altitude drawn to one of them equal respectively to two sides and the corresponding altitude of the other, the triangles are congruent. Suggestion. Read the note following § 77. Ex. 37. Construct a pattern for the pointed end of a belt, assuming that the belt material is 2 in. wide, and that the point is to project 1 in. beyond the square end of the belt. Ex. 38. Draw any straight line of indefinite length and select two points not in it. Find the point in the line which is equidistant from the two given points. Ex. 39. Find a point in one side of a triangle which is equidistant from the other two sides of the triangle. Ex. 40. Prove that either exterior angle at the base of an isosceles triangle is equal to the sum of a right angle and one half the vertical angle. Ex. 41. If from the vertex of one of the equal angles of an isosceles triangle a perpendicular be drawn to the opposite side, it makes with the base an angle equal to one half the vertical angle of the triangle. A Suggestion.- Construct the bisector of C. Ex. 42. ▲ ABC is an equilateral triangle. meets AC at P; CM, the bisector of exterior extended at M. MN is perpendicular to CR. Prove MN = BP. B BP, the bisector of ▲ B, angle ACR, meets BP Ex. 43. If the equal sides of an isosceles triangle be extended beyond the base, the bisectors of the exterior angles so formed form with the base another isosceles triangle. Ex. 44. If A ABC and ▲ ABD are two tri angles on the same base and on the same side of = it, such that AC BD and AD = BC, and AD and BC intersect at O, then ▲ OAB is isosceles. Ex. 45. If CD is the bisector of C of ▲ ABC, and DF be drawn parallel to AC meeting BC at E and the bisector of the angle exterior to Cat F, prove DE = EF. Suggestion. - Compare DE and EF with EC. Ex. 46. If equiangular triangles be constructed upon the sides of any triangle, the lines drawn from their outer vertices to the opposite vertices of the given triangle are equal. Suggestion. Recall § 124. Ex. 47. If AC be drawn from the vertex of the right angle to the hypotenuse of right ▲ BCD so as to make ZACD ZD, it bisects the hypotenuse. = Suggestion. - Prove LB LACB by § 109 and § 37. Ex. 48. If the angle at the vertex of isosceles ▲ ABC is equal to twice the sum of the equal angles B and C, and if CD is perpendicular to BC, meeting BA extended at D, prove ▲ ACD is equilateral. - B D A B. Suggestion. Determine the number of degrees in each angle of ▲ ABC. Ex. 49. If the bisectors of the equal angles of an isosceles triangle meet the equal sides at D and E respectively, prove that DE is parallel to the base of the triangle. Suggestions.-1. Compare LCED + A E CDE with A+B (§ 106). 2. Is CED = LCDE? 3. Is LCED = LA? B Ex. 50. Prove that two parallelograms are congruent if two sides and the included angle of one are equal respectively to two sides and the included angle of the other. Suggestion. Prove by superposition. Recall § 132. Ex. 51. Prove that the sum of the perpendiculars drawn from any point within an equilateral triangle to the sides of the triangle is equal to the altitude of the triangle. Prove OR+OF+ OD = BG. Suggestions. -1. Let KM be || AC and KE 1 AB. Ex. 52. An ironing board is supported on each side as shown in the adjoining figure. If AO = OB and DO = OC, prove that AC is always parallel to the floor DB. Ex. 53. What angle is formed by the bisectors of two consecutive angles of: (a) a rectangle? (b) an equilateral triangle? (c) a parallelogram? Ex. 54. Prove that the bisectors of the interior angles of a parallelogram form a rectangle. A B C F G H D Ex. 55. Construct a rhombus whose sides are each 3 in. and whose acute angles are each 45°. Draw and measure its diagonals. Ex. 56. Construct a rhombus, having given one side and one diagonal. Ex. 57. Prove that the two altitudes of a rhombus are equal. Ex. 58. If on the diagonal BD of square ABCD a distance BE is taken equal to AB, and if EF is drawn perpendicular to BD meeting AD at F, then AF EF = ED. Suggestion.- What kind of angle is angle EDF? Ex. 59. If AD and BD are the bisectors of the exterior angles at the ends of the hypotenuse AB of right triangle ABC, and DE and DF are perpendicular respectively to CA and CB extended, prove CEDF is a square. Suggestion. Recall § 143. Prove DE Ex. 60. = DF, using § 120, I. Prove that the bisectors of the angles of B a rectangle form a square. Suggestions. F E -1. Make a plan based upon § 143. G Ex. 61. If the non-parallel sides of an isosceles trapezoid are extended until they meet, they form with the base an isosceles triangle. Ex. 62. If the line joining the mid-points of the bases of a trapezoid is perpendicular to the bases, the trapezoid is isosceles. Ex. 63. If the bisectors of the interior angles of a trapezoid do not meet at a point, they form a quadrilateral, two of whose angles are right angles. Suggestion.- Prove FEH and FGH are right A4 angles. Ex. 64. If D is the mid-point of side AC of isosceles ▲ ABC, and DE is perpendicular to base BC, then EC is BC. Suggestion. -Draw DF parallel to AB. |