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§ 53 (two oblique lines drawn from the same point in a I, cutting off equal dis
tances from the foot of the I, are equal).
§ 18 (a straight line is the shortest distance between two points).
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 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 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. A E
Let B A (fig. 1) be perpendicular to C D at the point B.
We are to prove B A the only perpendicular to C D at the point B.
If it be possible, let B E be another line I to C D at B.
Ax. 1. 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 I to C D at B. Let A B (fig. 2) be perpendicular to C D from the point A.
We are to prove A B the only I to CD from the point A.
If it be possible, let A E be another line drawn from A I to C D.
Conceive 2 A E B to be moved to the right until the vertex E falls on B, the side E B continuing in the line C D.
Then the line E A will take the position BF.
Now if A E be I to C D, B F is I to C D, and there will be two Is to C D at the point B; which is impossible.
In like manner, it may be shown that no other line but . A B is I to C D 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.
If a straight line E F cut two other straight lines AB and C D, it makes with those lines eight angles, to which particular names are given.
The angles 1, 4, 6, 7 are called Interior 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 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.
Let the adjacent angles LOCA + LOCB= 2 rt. 6.
We are to prove A C and C B in the same straight line.
(being sup.-adj. 4).
..LOCA+20C F= LOCA + LOCB. Ax. 1. Take away from each of these equals the common 20CA. Then
LOCF= LOC 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.
Let A B be the given straight line, C the given point,
and Co the perpendicular.
We are to prove CO < any other line drawn from C to A B, as CF. Produce C O to E, making 0 E=CO.
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 0 C = 0 E by cons.).
$ 18 (having their extremities in the same points).
..CF + FE= 2 CF,
CO + O E<CF + FE,
2 CO< 2 CF.
Q. E. D.