THEOREMS AND DEMONSTRATIONS.

59. A geometric proposition is a statement affirming certain properties of geometric figures.

Thus: "Two points determine a straight line" and "The base angles of an isosceles triangle are equal" are geometric propositions. A proposition is proved or demonstrated when it is shown. to follow from other known propositions.

A theorem is a proposition which is to be proved. The argument used in establishing a theorem is called a proof.

60. In every mathematical science some propositions must be left unproved, since every proof depends upon other propositions which in turn require proof. Propositions which for this reason are left unproved are called axioms.

While axioms for geometry may be chosen in many different ways, it is customary to select such simple propositions as are evident on mere statement.

61. Among the axioms thus far used are the following: Axioms. I. A figure may be moved about in space without changing its shape or size. See § 26.

II. Through two points one and only one straight line can be drawn. See §§ 8, 32.

III. The shortest distance between two points is measured along the straight line-segment connecting them.

Thus one side of a triangle is less than the sum of the other two. IV. If each of two figures is congruent to the same figure, they are congruent to each other. See §§ 28, 40.

V. If a, b, c, d are line-segments (or angles) such that ab and c=d, then a+c=b+d and a-c-b-d. In the latter case we suppose a >c, b>d. See §§ 10, 39.

VI. If a and b are line-segments (or angles) such that a = b, then a × n = b x n and a ÷ n = b÷n; and if a>b, then a × n>b×n and a÷n>b÷n, n being a positive integer. See §§ 10, 39.

NOTE.

An equality or an inequality may be read from left to right or from right to left. Thus, a >b may also be read b<a.

Other axioms are given in §§ 82, 96, 119, and in Chapter

VII.

Certain other simple propositions may be assumed at present without detailed proof. These are called preliminary theorems.

PRELIMINARY THEOREMS.

62. Two distinct lines can meet in only one point.

For if they have two points in common, then by Ax. II they are the same line.

63. All straight angles are equal. § 20, Ex. 1.

64. All right angles are equal. See Ax. VI.

65. Every line-segment has one and only one middle point.

See § 51, where the middle point is found by construction. 66. Every angle has one and only one bisector. See § 48, where the bisector is constructed.

67. One and only one perpendicular can be drawn to a line through a point whether that point is on the line or not. See § 20, Exs. 4, 5; § 49, Ex. 4; § 54.

68. The sum of all the angles about a point in a straight line and on one side of it is two right angles.

69. The sum of all the angles about a point in a plane is four right angles.

In §§ 68, 69 no side of one angle is to lie inside another.

70. Definitions.

Two angles are said

to be complementary if their sum is one right angle. Each is then called the complement of the other.

Thus, a and b are complementary angles. Two angles are said to be supplementary if their sum is two right angles. Each is then said to be the supplement of the other.

Thus, 1 and 2 are supplementary angles. Two angles are called vertical angles if the sides of one are prolongations of the sides of the other.

A.

Thus, 21 and 23 are vertical angles, and also ≤2 and 24.

1. What is the complement of 45°? the supplement?

2. If the supplement of an angle is 140°, find its complement. 3. If the complement of an angle is 21o, find its supplement. 4. Find the supplement of the complement of 30°.

5. Find the angle whose supplement is five times its complement. 6. Find the angle whose supplement is n times its complement. 7. Find an angle whose complement plus its supplement is 110°.

8. If in the first figure b = 22a, and c = Za + 2b, find each angle.

9. If in the second figure 2b=12a, 2c=2a+2b, and Zd=64a, find each angle.

10. If in the third figure 2b = Le, Lc = La + Zb, 2 d = 2 ≤ b, and Zed, find each angle.

PRELIMINARY THEOREMS.

72. Angles which are complements of the same angle or of equal angles are equal.

For they are the remainders when the given equal angles are subtracted from equal right angles. Ax. V.

73. Angles which are supplements of the same angle or of equal angles are equal.

For they are the remainders when the given equal angles are subtracted from equal straight angles.

74. Vertical angles are equal.

They are supplements of the same angle.

75. If two adjacent angles are supplementary, their exterior sides are in the same straight line.

For the two angles together form a straight angle.

76. If two adjacent angles have their exterior sides in the same straight line, they are supplementary. For a straight angle is equal to two right angles.

1. Prove that if one of the four angles formed by two intersecting straight lines is a right angle, then all are right angles.

2. Show that the rays bisecting two complementary adjacent angles form an angle of 45°.

3. Find the angle formed by the rays bisecting two supplementary adjacent angles. Prove.

4. Find the angle formed by the rays bisecting two vertical angles. Prove.

5. The sum of two adjacent angles is 74°. Find the angle formed by their bisectors.

6. The angle formed by the bisectors of two adjacent angles is 37° 18'. Find the sum of the adjacent angles.

ON THE NATURE OF A DEMONSTRATION.

78. A theorem consists of two distinct parts, hypothesis and conclusion.

In a geometrical theorem, the hypothesis specifies certain properties which the figures in question are assumed to possess. The conclusion asserts that certain other properties belong to the figures whenever the assumed properties are present.

The hypothesis and conclusion are often intermingled in a single statement, in which case they should be explicitly separated before making the proof.

For example, in the theorem of § 37, The angles opposite the equal sides of an isosceles triangle are equal, the hypothesis is, "Two sides of a triangle are equal," and the conclusion is, "The angles opposite them are equal."

79. If the hypothesis consists of several parts, these should be tabulated and then checked off as the demonstration proceeds. If the theorem is properly stated, each part of the hypothesis will be used in the proof.

For instance, in the theorem of § 32, the hypothesis is: AB = A'B', AC=A'C', and ZA = ▲ A'; and the conclusion is: ▲ ABC≈▲ A'B'C'. It will be found on examining the proof that each part of the hypothesis is needed and used in the course of the demonstration.

If the conclusion could be proved without using every part of the hypothesis, then the parts not used should be omitted from the hypothesis in the statement of the theorem.

80. In the proof of a theorem no conclusion should be taken for granted simply from the appearance of the figure. Each step in a proof should be based upon a definition, an axiom, or a theorem previously proved.

It will then follow that the theorem is as certainly true as are the simple, unproved propositions with which we start, and upon which our argument is based.