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40. DEF. An Axiom is a truth which is admitted without demonstration.

41. DEF. A Postulate is a problem which is admitted to be possible.

42. DEF. A Proposition is either a theorem or a problem.

43. DEF. A Corollary is a truth easily deduced from the proposition to which it is attached.

44. DEF. A Scholium is a remark upon some particular feature of a proposition.

45. DEF. An Hypothesis is a supposition made in the enunciation of a proposition, or in the course of a demonstration.

46. Axioms. 1. Things which are equal to the same thing are equal to each

other. 2. When equals are added to equals the wholes are equal. 3. When equals are taken from equals the remainders are equal. 4. When equals are added to unequals the wholes are unequal. 5. When equals are taken from unequals the remainders are

unequal. 6. Things which are double the same thing, or equal things,

are equal to each other. 7. Things which are halves of the same thing, or of equal

things, are equal to each other. 8. The whole is greater than any of its parts. 9. Every whole is equal to all its parts taken together.

47. POSTULATES.. Let it be granted — 1. That a straight line can be drawn from any one point to any

other point. 2. That a straight line can be produced to any distance, or can

be terminated at any point. 3. That the circumference of a circle can be described about any

centre, at any distance from that centre.

48. SYMBOLS AND ABBREVIATIONS.

is therefore. ·

Post. postulate. = is (or are) equal to. Def. definition. z angle.

Ax. axiom. & angles.

Hyp. hypothesis. A triangle.

Cor. corollary. A triangles.

Q. E. D. quod erat demonstran|| parallel.

dum. o parallelogram

Q. E. F. quod erat faciendum. S parallelograms. Adj. adjacent. I perpendicular. Ext.-int. exterior-interior.

Is perpendiculars. Alt.-int. alternate-interior. rt. Z right angle.

Iden. identical. rt. As right angles.

Cons. construction. > is (or are) greater than. Sup. supplementary.

< is (or are) less than. Sup. adj. supplementary-adjart. A right triangle.

cent. rt. A right triangles.

Ex. exercise. O circle.

Ill. illustration. © circles. + increased by. – diminished by. X multiplied by. ; divided by.

ON PERPENDICULAR AND OBLIQUE LINES.

PROPOSITION I. THEOREM. 49. When one straight line crosses another straight line the vertical angles are equal.

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Let line 0 P cross A B at C.

We are to prove 20CB= 2 ACP.
LOCA + LOCB= 2 rt. 4, § 34

(being sup.-adj. 6).
LOCA+ZACP= 2 rt. £,

(being sup.-adj.). ::LOCA + LOCB=LOCA + LACP. Ax. 1.

Take away from each of these equals the common ZOCA.
Then

.LOCB=LACP.

In like manner we may prove

LACO= L PC B.

Q. E. D.

50. COROLLARY. If two straight lines cut one another, the four angles which they make at the point of intersection are together equal to four right angles.

PROPOSITION II. THEOREM.

51. When the sum of two adjacent angles is equal to two right angles, their exterior sides form one and the same straight line.

----F

B

Let the adjacent angles 2 OCA + 2OC B= 2 rt. E.

We are to prove A C and C B in the same straight line.
Suppose C F to be in the same straight line with AC.
Then ZOCA + LOCF= 2 rt. . $ 34

(being sup.-adj. €).
But ZOCA + LOCB= 2 rt. 6. Hyp.

..LOCA+20CF=LOCA+ LOC B. Ax. 1.

Take away from each of these equals the common 20C A.
Then

ZOCF=2OC B. .:. C B and C F coincide, and cannot form two lines as represented in the figure. .. A C and C B are in the same straight line.

Q. E. D.

PROPOSITION III. THEOREM. 52. A perpendicular measures the shortest distance from a point to a straight line.

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Let A B be the given straight line, C the given point,

and Co the perpendicular.

We are to prove CO< any other line, as C F.
Produce C O to E, making 0 E=C0.

Draw E F. On A B as an axis, fold over O C F until it comes into the plane of O EF.

The line 0 C will take the direction of O E,
(since 2COF= L E O F, each being a rt. 2).
The point C will fall upon the point E,

(since OC=0 E by cons.).
... line C F= line F E,

§ 18 (having their extremities in the same points).

..CF + FE= 2 C F,
CO+O E= 2 CO.

Cons.
But
CO+OE<CF + FE,

§ 18
(a straight line is the shortest distance between two points).
Substitute 2 CO for CO + O E, .
and 2 C F for C F + FE; then we have

2 CO< 2 C F ..CO< CF.

Q. E. D.

and

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