PROPOSITION VII. THEOREM.

57. Two equal oblique lines, drawn from the same point in a perpendicular, cut off equal distances from the foot of the perpendicular.

Let C F be the perpendicular, and C E and C K be two equal oblique lines drawn from the point C.

Fold over CFA on CF as an axis, until it comes into the plane of CFB.

The line FE will take the direction FK,

(<CFE=LCF K, each being a rt. 4).

Then the point E must fall upon the point K;

otherwise one of these oblique lines must be more remote from the,

and .. greater than the other; which is contrary to the hypothesis.

.. FEF K.

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PROPOSITION VIII. THEOREM.

58. If at the middle point of a straight line a perpendicular be erected,

I. Any point in the perpendicular is at equal distances from the extremities of the straight line.

II. Any point without the perpendicular is at unequal distances from the extremities of the straight line.

Let PR be a perpendicular erected at the middle of the straight line A B, O any point in PR, and C any point without PR.

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

We are to prove CA and C B unequal.

One of these lines, as CA, will intersect the 1.

From D, the point of intersection, draw D B.

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

CBCD+DB,

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

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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.

twa

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 beg of a right angle, what is its complement? 4. If an angle be 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

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.

Let BA (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 B E be another line to C D at B.

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 AB (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 A E be another line drawn from A
to CD.

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 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. will wai Parallel lines lie in the same direction, when they are on ex par prizinci

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.

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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.

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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