28. Cut out four equal squares and place them around a common point so that an angle of each shall have its

vertex at the point.

around the point?

Will they form a continuous surface

B

29. Make six equal equilateral triangles and place them around a common point, as E. They form a regular hexagon. How many degrees are there in each angle formed at E? How many degrees are there in angle ABC? BCD? In each angle of the hexagon? Give reasons.

A

30. How many sides has the polygon ADCB? Which of its sides are parallel?

E

NOTE. A quadrilateral which has only two of its sides parallel is called a Trapezoid.

31. Draw a trapezoid and show into how many triangles it can be divided by one line.

32. If each of the sides of the equilateral triangles in Ex. 29 is 8 centimeters, what is the perimeter of the trapezoid BGDC?

33. Inclose a surface by three unequal lines, and name the plane figure thus formed.

NOTE. A triangle whose sides are all unequal is called a Scalene Triangle.

34. Construct a triangle whose sides are 5 inches, 6 inches, and 7 inches, using the 7-inch line as the base.

SUGGESTION. With the extremities of the base as centers describe arcs, with radii respectively 5 inches and 6 inches, and join the point of intersection with the extremities of the base.

35. Construct a triangle whose sides are 8 centimeters, 7 centimeters, and 6 centimeters.

36. Construct a triangle whose sides are 9 inches, 3 inches, and 4 inches. Explain.

37. What is the perimeter of the scalene triangle ABC, in which AB is 12 inches, BC is 2 inches longer than AB, and AC is 3 inches longer than BC?

38. The perimeter of a scalene triangle is 47 inches; one side is 11 inches and another side is 11 times as long. Find the third side.

39. The triangle ABC, whose perimeter is 54 inches, has the side AB 7 inches longer than the side BC, and the side BC 10 inches longer than the side AC. Find each side.

SUGGESTION. Let x AC.

40. The scalene triangle ABC has the side AB 12 inches longer than the side AC, and the side AC 8 inches longer than the side BC. The perimeter is 73 inches. Find each side.

41. The side XY of the scalene triangle XYZ is 11 inches longer than the side YZ, and the side XZ is 17 inches longer than the side YZ. The perimeter is 88 inches. Find each side.

42. The side DE of the scalene triangle DEF, the perimeter of which is 65 inches, lacks 8 inches of being twice as long as the side EF. The side DF lacks 17 inches of being 3 times as long as EF. Find the length of each side.

43. Quote the geometric principle which tells how many degrees there are in the sum

of the angles a, b, and c, AB being a straight line.

A

B

44. Draw and cut out a triangle, dividing each of its angles by straight lines. Cut off two of the corners. Place them beside the third corner.

It will be seen that all the angles will have their vertex at a common point, and will lie on the same side of a straight line. How many degrees are there in the sum of all the angles?

PRINCIPLE 19.-The sum of the angles of any triangle is equal to two right angles, or 180 degrees.

45. Draw different kinds of triangles, and repeat the process given in Ex. 44 illustrating Prin. 19.

46. How many degrees are there in the vertical angle of an isosceles triangle whose base angles are each 80° ? 47. How many degrees are there in each base angle of an isosceles triangle whose vertical angle is 50° ?

48. The angle a of a triangle is 80°, and angle b is 3 times angle c. Find angle b and angle c. Let x = ?

49. Make a right angle, and join the extremities of the lines. How many degrees are there in the sum of the two angles that are not right angles?

NOTE. — A triangle which has a right angle is a Right Triangle. The side opposite the right angle is called the Hypotenuse.

50. How many degrees are there in each angle of an isosceles triangle whose vertical angle is equal to the sum of the base angles?

51. Draw a right-angled isosceles triangle having its equal sides each 5 inches long, and show how many degrees there are in each of the complementary angles. QUERY. When is one angle complementary to another?

52. Draw a right-angled isosceles triangle having its equal sides each 10 inches long, and show how many degrees there are in each of the complementary angles.

53. In the scalene triangle ABC the angle A is a right angle, and the angle B is 4 times the angle C. How many degrees are there in each of the complementary angles?

54. Find each of the complementary angles in a right triangle, in which one acute angle is 5 times the other.

55. Construct an angle of 60° at one extremity of a line and one of 70° at the other extremity. Prolong the lines until they meet. How many degrees are there in the third angle?

NOTE. A triangle in which all the angles are acute is an Acuteangled Triangle.

56. Draw an acute-angled triangle in which one angle is 80°.

57. Draw an obtuse angle, and join the ends of the lines which form it. What kind of triangle is thus formed?

NOTE. A triangle which has an obtuse angle is an Obtuse-angled Triangle.

58. Can you draw an acute-angled triangle in which the sum of any two angles is less than a right angle?

59. To what is the sum of the oblique angles of a right triangle equal?

60. Try to draw an obtuse-angled triangle in which the sum of the acute angles is greater than a right angle. Explain.

61. BD is perpendicular to AC, one of the legs of the isosceles triangle ABC, whose vertical angle is 40°. How many degrees are there in each of the angles x, y, and z?

62. BD bisects the vertical angle of the isosceles triangle ABC, a base angle of which is 65°. How many degrees are there in angle ? In angle y? In angle m? In angle k? What is the relative position of BD and AC?

63. AD is a bisector of a base angle of the isosceles triangle whose vertical angle B is 48°. How many degrees are there in angle ADB?

64. In the isosceles triangle ABC the vertical angle is 38°. AE bisects one base angle. DC bisects the other. Angles of how many degrees are formed at F?

65. Angles of how many degrees are

A

A

B

A

B

mk

B

B

D

D

E

formed by the intersection of the bisector of the vertical angle and the bisector of a base angle in the triangle whose base angles are each 20° ?

66. What angles are formed by the intersection of the lines drawn from the vertices of the base angles perpendicular to the legs of an isosceles triangle whose vertical angle is 30° ?