10. Doubles or halves of unequals are unequal in the same sense. 11. If the first of three magnitudes is greater than the second, and the second is greater than the third, the first is greater than the third. 12. A straight line is the shortest line that can be drawn between two points.

13. A geometrical figure may be moved from one position to another without any change in form or magnitude.

32. A postulate is something required to be done, the possibility of which is admitted as evident.

33. POSTULATES.

1. It is possible to draw a straight line from any point to any other point.

2. It is possible to extend (prolong or produce) a straight line indefinitely, or to terminate it at any point.

34. A geometric proof or demonstration is a logical course of reasoning by which a truth becomes evident.

35. A theorem is a statement that requires proof.

In the case of the preliminary theorems which follow, the proof is very simple; but as these theorems are not selfevident they cannot be classified with the axioms.

A corollary is a truth immediately evident, or readily established, from some other truth or truths.

EXERCISE 1. Draw an ABC. In ABC draw line BD.

What does / ABD + 2 DBC = ?

What does LABC - LABD = ?

Ex. 2. In a rt. Z ABC draw line BD.

If / ABD=25°, how many degrees are there in DBC?

How many degrees are there in the complement of an angle of 38°? How many degrees are there in the supplement?

Ex. 3. Draw a straight line AB and take a point X on it.

What line does AX + BX = ?

What line does AB - BX ?

=

Ex. 4. Draw a straight line AB and prolong it to X so that BX = AB. Prolong it so that AX = AB.

BOOK I

ANGLES, LINES, RECTILINEAR FIGURES

PRELIMINARY THEOREMS

36. A right angle is equal to half a straight angle.
Because of the definition of a right angle. (See 16.)

37. A straight angle is equal to two right angles. (See 36.)
38. Two straight lines can intersect in only one point.

Because they would coincide entirely if they had two common points. (See 5.)

39. Only one straight line can be drawn between two points. (See 5.)

40. A definite (limited or finite) straight line can have only one midpoint.

Because the halves of a line are equal.

41. All straight angles are equal.

Because they can be made to coincide. (See 28 and Ax. 13.)

42. All right angles are equal.

They are halves of straight angles (36), and hence equal (Ax. 3).

43. Only one perpendicular to a line can be drawn from a point in the line.

Because the right angles would not be equal if there were two perpendiculars; and all right angles are equal. (See 42.)

44. If two adjacent angles have their exterior sides in a straight

Because their sum is two rt. 4 (20); or a straight (37). Hence the exterior sides are in the same straight

line (18).

46. The sum of all the angles on one side of a straight line at a point equals two right angles.

(See Ax. 4 and 37.)

47. The sum of all the angles about

a point in a plane is equal to four

right angles. (See 46.)

48. Angles that have the same complement are equal. Or, complements of the same angle, or of equal angles, are equal.

Because equal angles subtracted from equal right angles leave equals. (See Ax. 2.)

49. Angles that have the same supplement are equal. Or, supplements of the same angle, or of equal angles, are equal. (See Ax. 2.)

50. If two angles are equal and supplementary, they are right angles.

Because each is half a straight; hence each is a rt. Z. (See 36.)

NOTE. A single number, given as a reference, always signifies the truth stated in that paragraph and is usually the statement in full face type only. In reciting or writing the demonstrations the pupil should quote the correct reason for each statement, and not give the number of its paragraph. [Consult model demonstrations on page 24.]

ZBOL is the supplement of Z MOB. (Why?) (See 44.) .. ▲ AOM=/ BOL. (Why ?) (See 49.)

AOL and BOM are a pair of vertical angles. equal in precisely the same manner. If ZAOL = are there in the other ?

These may be proved 80°, how many degrees

52. THEOREM. Two triangles are equal if two sides and the included angle of one are equal respectively to two sides and the included angle of the other.

Proof : Place the ▲ ABC upon the ▲ RST so that ▲ ▲ coincides with its equal, R; then AB will fall upon RS and point B upon s. (It is given that AB = RS.) AC will fall upon RT and point Cupon T. (It is given that AC RT.) .. BC will coincide with ST.

(Why?) (See 39.)

Hence, the triangles coincide in every respect and are

equal (28).

53. THEOREM. Two right triangles are equal if the two legs of one are equal respectively to the two legs of the other.

54. THEOREM. Two triangles are equal if a side and the two angles adjoining it in the one are equal respectively to a side and the two angles adjoining it in the other.

B

K

Given: ABCD and JKL; BC= JK; LB = LJ; < C = ZK.

To Prove: ▲ BCD = ▲ JKL.

Proof: Place ▲ BCD upon ▲ JKL so that B coincides with its equal, ▲ J, BC falling on JK.

Point C will fall on K. (It is given that BC= JK.)

BD will fall on JL.

CD will fall on KL.

(Because

≤

B is given = 2 J.) (Because C is given K.) = <

Then point D which falls on both the lines JL and KL will

fall at their intersection, L. (Why?) (See 38.)

.. the are. (Why?) (See 28.)