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(two oblique lines drawn from the same point in a 1, cutting off equal distances from the foot of the 1, are equal).

CB < CD+DB,

(a straight line is the shortest distance between two points).

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$ 18

Ax. 9.

Q. E. D,

59. The Locus of a point is a line, straight or curved, containing all the points which possess a common property.

Thus, the perpendicular erected at the middle of a straight line is the locus of all points equally distant from the extremities of that straight line.

60. SCHOLIUM. Since two points determine the position of a straight line, two points equally distant from the extremities of a straight line determine the perpendicular at the middle point of that line.

Ex. 1. If an angle be a right angle, what is its complement? 2. If an angle be a right angle, what is its supplement ?

3. If an angle be

4. If an angle be

of a right angle, what is its complement?

of a right angle, what is its supplement? 5. Show that the bisectors of two vertical angles form one and the same straight line.

6. Show that the two straight lines which bisect the two pairs of vertical angles are perpendicular to each other.

PROPOSITION IX. THEOREM.

61. At a point in a straight line only one perpendicular to that line can be drawn; and from a point without a straight line only one perpendicular to that line can be drawn.

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Let B A (fig. 1) be perpendicular to C D at the point B. We are to prove BA the only perpendicular to CD at the point B.

If it be possible, let BE be another line to C D at B.

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That is, a part is equal to the whole; which is impossible. In like manner it may be shown that no other line but BA is to CD at B.

Let A B (fig. 2) be perpendicular to CD from the point A.
We are to prove A B the only to CD from the point A.
If it be possible, let AE be another line drawn from A
to C D.

Conceive AEB to be moved to the right until the vertex E falls on B, the side E B continuing in the line CD. Then the line E A will take the position B F.

Now if A E be 1 to CD, BF is 1 to CD, and there will be two to CD at the point B; which is impossible.

In like manner, it may be shown that no other line but AB is L to CD from A.

Q. E. D.

62. COROLLARY. Two lines in the same plane perpendicular to the same straight line have the same direction; otherwise they would meet (§ 22), and we should have two perpendicular lines drawn from their point of meeting to the same line; which is impossible.

ON PARALLEL LINES.

63. Parallel Lines are straight lines which lie in the same plane and have the same direction, or opposite directions.

Parallel lines lie in the same direction, when they are on the same side of the straight line joining their origins.

Parallel lines lie in opposite directions, when they are on opposite sides of the straight line joining their origins.

64. Two parallel lines cannot meet.

$21 65. Two lines in the same plane perpendicular to a given line have the same direction (§ 62), and are therefore parallel. 66. Through a given point only one line can be drawn parallel to a given line. § 18

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If a straight line EF cut two other straight lines A B and CD, it makes with those lines eight angles, to which particular names are given.

The angles 1, 4, 6, 7 are called Interior angles.

The angles 2, 3, 5, 8 are called Exterior angles.

The pairs of angles 1 and 7, 4 and 6 are called Alternateinterior angles.

The pairs of angles 2 and 8, 3 and 5 are called Alternateexterior angles.

The pairs of angles 1 and 5, 2 and 6, 4 and 8, 3 and 7 are called Exterior-interior angles.

PROPOSITION X. THEOREM.

67. If a straight line be perpendicular to one of two parallel lines, it is perpendicular to the other.

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Let A B and E F be two parallel lines, and let HK be

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(Two lines in the same plane to a given line are parallel).

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$ 65

Hyp.

§ 66

(Through the same point only one line can be drawn || to a given line).

that is

.. EF is to HK,

HK is to EF.

Q. E. D.

PROPOSITION XI. THEOREM.

68. If two parallel straight lines be cut by a third straight line the alternate-interior angles are equal.

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Let E F and GH be two parallel straight lines cut by

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Through O, the middle point of B C, draw A D1 to G H.

Then

AD is likewise to EF,

(a straight line to one of two lls is to the other),

that is, CD and B A are both 1 to A D.

$67

Apply figure COD to figure BOA so that OD shall fall

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(from a point without a straight line only one to that line can be drawn). .. ZO CD coincides with Z OBA, and is equal to it.

Q. E. D.

SCHOLIUM. By the converse of a proposition is meant a proposition which has the hypothesis of the first as conclusion and the conclusion of the first as hypothesis. The converse of a truth is not necessarily true. Thus, parallel lines never meet; its converse, lines which never meet are parallel, is not true unless the lines lie in the same plane.

NOTE. The converse of many propositions will be omitted, but their statement and demonstration should be required as an important exercise for the student.

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