47. The enunciation of a theorem may be separated into the following parts:

1. The things given, or granted, called the Data (singular datum).

2. A statement of what is to be proved, called the Conclusion. The term Hypothesis may be used instead of the term data. A supposition made in the course of a demonstration is also called an Hypothesis.

48. In proofs, or demonstrations, only definitions, axioms, and propositions which have been proved can be employed to establish the truth of the proposition.

AXIOMS

49. 1. Things which are equal to the same thing are equal to each other.

2. If equals are added to equals, the sums are equal.

3. If equals are taken from equals, the remainders are equal.

4. If equals are added to unequals, the sums are unequal.

5. If equals are taken from unequals, the remainders are unequal.

6. Things which are doubles of equal things are equal.

7. Things which are halves of equal things are equal. 8. The whole is greater than any of its parts.

9. The whole is equal to the sum of all its parts.

10. A straight line is the shortest distance between two points. 11. If two straight lines coincide in two points, they will coincide throughout their whole extent, and form one and the same straight line.

12. Between the same two points but one straight line can be drawn.

POSTULATES

50. 1. A straight line may be produced indefinitely.

2. A straight line may be drawn from any point to any other point.

3. On the greater of two straight lines a part can be laid off equal to the less.

4. A figure can be moved unaltered to a new position.

The letters Q.E.D. are placed at the end of a proof; they are the initial letters of the Latin words quod erat demonstrandum, meaning which was to be proved.

The letters Q.E. F. are placed at the end of a solution of a problem for quod erat faciendum, meaning which was to be done.

PLANE GEOMETRY

BOOK I

LINES AND RECTILINEAR FIGURES

Proposition I

51. Draw a straight line and as many perpendiculars as possible to the line at one point. How many can be drawn? (§ 37 )

Theorem. At any point in a straight line one perpendicular to the line can be drawn, and only one.

Data: Any straight line, as AB, and

any point in that line, as o.

To prove that a perpendicular to AB can be drawn at the point 0, and that only one can be drawn.

A

Proof. Suppose a line DO to rotate about the point o as a pivot, from the position BO to 40.

B

As DO rotates from the position BO toward the position 40, the angle DOB will, at first, be smaller than the angle DOA.

AS DO continues to rotate, the angle DOB will increase continuously, and will eventually become larger than angle DOA.

Therefore, since angle DOB is at first smaller than angle DOA, and afterwards larger than angle DOA, there must be one position of DO, as, for example, Co, in which the two angles are equal. By 26, co is then perpendicular to AB.

Since there is but one position in which the line DO makes equal angles with the line 4B, there can be but one perpendicular. Therefore, at any point in a straight line one perpendicular to the line can be drawn, and only one.

Q.E.D.

Proposition II

52. 1. Draw two lines intersecting so as to form a right angle. How does each of the other angles formed compare in size with a right angle? How do right angles compare in size? How do straight angles compare? (§ 37 )

2. Draw two equal angles and their complements. How do their complements compare in size? How do their supplements compare? (§ 37) Theorem. All right angles are equal.

Proof. Suppose that ▲ DEF is placed upon ▲ ABC in such a way that the point E falls upon the point B and the line ED takes the direction of the line BA.

Since by § 26, BC is perpendicular to BA and EF is perpendicular to ED and on the same side of the line,

line EF must take the same direction as line BC,

for otherwise there would be two perpendiculars to BA at the point B and by § 51, this is impossible.

Consequently, the line EF falls upon the line BC,

Hence, § 36,

and

DEF coincides with Z ABC.
ABC and DEF are equal.

Therefore, all right angles are equal.

53. Cor. I. All straight angles are equal.

Q.E.D.

54. Cor. II. The complements of equal angles are equal, also

the supplements of equal angles are equal.

Ex. 1. Find the complement of an angle of 15°; 27°; 35°; 49°.

Ex. 2. Find the supplement of an angle of 38°; 96°; 114°.

Ex. 3. The complement of an angle is 63°.

What is the angle?
What is the angle ?

Ex. 4. The supplement of an angle is 103°. Ex. 5. Find the complement of the supplement of an angle of 165°; 140°; 122°; 113°; 108°; 99°.

Ex. 6. Find the supplement of the complement of an angle of 48°; 84°; 27°; 16°; 31°; 54°; 39°.

Proposition III

55. 1. Draw a straight line, and another meeting it. How does the sum of the adjacent angles thus formed compare with a right angle? With a straight angle? (§ 37 )

2. Draw a straight line, and from any point in it draw several lines extending in different directions. How does the sum of the consecutive angles formed on one side of the line compare with a right angle? With a straight angle? How does the sum of the consecutive angles on both sides of the line compare with a right angle? With a straight angle? (§ 37)

Theorem. If one straight line meets another straight line, the sum of the adjacent angles is equal to two right angles.

Data: Any straight line, as AB, and any other straight line, as Co, meeting it in the point 0.

To prove the sum of the adjacent angles, 40C and t, equal to two right A angles.

Proof. When co is perpendicular to AB,

by § 26,

and

each of the AOC and t is a rt. 2,

their sum is two rt. .

When co is not perpendicular to AB,

draw DO perpendicular to AB at the point 0.

Therefore, if one straight line meets another straight line, the sum of the adjacent angles is equal to two right angles.

Q.E.D.