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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).
we off equal
CB<CD + DB,
Substitute for D B its equal D A, then
CB C D + DA.
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 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; anil from a point without a straight line only one perpendicular to that line can be drawn.
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.
$ 26 But
Z ABD is a rt. Z.
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 C D from the point A.
If it be possible, let A E be another line drawn from A I to C D.
Conceive ZA E B to he 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 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 X. THEOREM.
67. If a straight line be perpendicular to one of two parallel lines, it is perpendicular to the other.
Let A B and EF be two parallel lines, and let H K be
perpendicular to A B.
We are to prove HKI to E F.
Through C draw MNI to H K.
Hyp. ..E F coincides with M N.
§ 66 (Through the same point only one line can be drawn II to a given line).
.. E F is I to HK,
H K is I to E F.
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.
Let E F and G H be two parallel straight lines cut by
the line BC.
Apply figure COD to figure BOA so that O D shall fall on 0 A. Then
OC will fall on O B,
point C will fall upon B,
(since 0 C = 0 B by construction). Then ICD will coincide with I BA,
§ 61 (from a point without a straight line only one I to that line can be drawn). ..ZOCD coincides with ZOBA, 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.