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CHAPTER IV

CONGRUENT TRIANGLES

The two basic methods are: *52. If two triangles have two sides and the included angle of the one equal respectively to two sides and the included angle of the other, the triangles are congruent.

*53. If two triangles have two angles and the included side of the one equal respectively to two angles and the included side of the other, the triangles are congruent.

54. What are the corresponding parts in Article 53? What can be said of the corresponding parts?

55. If two triangles have two angles and a side of the one equal respectively to two angles and the corresponding side of the other, the triangles are congruent.

56. What are the basic theorems of congruent triangles? How are the other theorems proved ?

57. What is a Right Triangle? Hypotenuse? Arm?

58. If two right angles have two arms of the one equal respectively to two arms of the other, the triangles are congruent.

59. If two right triangles have a side and an acute angle of the one equal respectively to the corresponding side and acute angle of the other, the triangles are congruent.

*60. If two sides of a triangle are equal, the opposite angles are equal.

*61. If two right triangles have the hypotenuse and an arm of the one equal respectively to the hypotenuse and an arm of the other, the triangles are congruent.

*62. If two triangles have three sides of the one equal respectively to three sides of the other, the triangles are congruent.

63. From Articles 58, 59, and 61 a general theorem for all right triangles can be made:

If two right triangles have a side and other part except the right angle of the one equal respectively to a corresponding side and other part except the right angle of the other, the triangles are congruent.

64. What is an Altitude of a triangle? A Median? An Angle-bisector?

65. Define Circle; Radius; Arc. *66. PROBLEM: Construct a triangle congruent to a given triangle.

SUMMARY

67. State four methods of proving triangles congruent, covering all cases.

What is the principal use of congruent triangles?

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1. If AB=AC and ZBAQ=2 CAO, prove ABAOYACAO.

2. If AB=AC and AO I BC, prove ABAO YA CAO.

8. If ZB=LC and AO I BC, prove ABAO YA CAO.

4. If AB=AC and BO=CO, prove ДВАО 2 ДСАО. . 5. If BO= CO and AO I BC, prove

B ДВАО ЯДСАО. .

6. If ZBAO= ZCAO and AO I BC, prove ABAO E A CAO.

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16. Illustrate by a figure:

a. A triangle with two sides equal.
b. A triangle with three sides equal.
c. The three altitudes of a triangle.
d. The three medians of a triangle.
e. The three angle-bisectors of a triangle.
f. In a triangle with two sides equal, the altitude, the median,

and the angle-bisector at the included vertex.
g. In a triangle with three sides equal, the altitude, the median,

the angle-bisector at any one vertex.

CHAPTER V

EQUALITY FROM CONGRUENT TRIANGLES

The basic method is:

The corresponding parts of congruent triangles are equal. (Art. 54.)

69. Define Scalene triangle.
70. Define Isosceles triangle.
71. Define Equilateral triangle.
72. Define Equiangular triangle.
Review Art. 60.

73. If three sides of a triangle are equal, the three angles are equal.

*74. If two angles of a triangle are equal, the opposite sides are equal.

75. If three angles of a triangle are equal, the three sides are equal.

*76. The perpendicular from the vertex to the base of an isosceles triangle:

1. Divides the triangle into two congruent triangles.

2. Bisects the angle at the vertex. *77. If equal lines are drawn from a point in a perpendicular to a base line, they cut off equal distances on the base line.

*78. If lines are drawn from a point in a perpendicular to a base line cutting off equal distances on the base line, they are equal. *79. PROBLEM: Construct an angle equal to a given angle. *80. PROBLEM: Bisect a given angle.

81. PROBLEM: Construct a triangle, having given two sides and the included angle.

82. Problem: Construct a triangle, having given two angles and the included side.

83. What should be memorized? What should not?

84. What is a Converse theorem? An Opposite theorem?

SUMMARY

85. State two methods of proving lines equal. State two new methods of proving angles even. State the commonest method for each.

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1. Any point in the bisector of the vertical angle of an isosceles triangle is equidistant from the ends of the base.

2. If the bisector of an angle of a triangle is perpendicular to the opposite side, the triangle is isosceles.

3. The median to the base of an isosceles triangle bisects the vertical angle.

4. If the median to one side of a triangle is perpendicular to that side, the triangle is isosceles.

5. The altitudes upon the sides of an isosceles triangle are equal.

6. If the altitudes upon two sides of a triangle are equal, the triangle is isosceles.

7. In an isosceles triangle, the lines drawn from the ends of the base to the middle of the sides (medians) are equal.

8. If two intersecting lines bisect each other, the opposite lines joining their ends are equal.

9. The bisectors of the base angles of an isosceles triangle are equal.

10. If D is in the middle point of side BC of triangle ABC, and BE and CF are lines from B and C perpendicular to AD, produced if necessary, then BE CF.

11. The perpendiculars drawn from the middle point of the base to the sides of an isosceles triangle are equal.

12. State and prove the converse of 11.

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