PROPOSITION III. THEOREM

112 Vertical angles are equal.

HYPOTHESIS. AB and CD are two intersecting lines.
CONCLUSION. ≤ m = ≤ n, and ≤ s = Lt.

17 If ≤1 = 38°, find ≤3, 24, 22.

18 If 3 is four times 1, find each angle.

19 The straight line which bisects one of two vertical angles bisects the other also. [Let HK bisect ZAOD; then HK bisects / COB. PROOF. 23=22, and 4 = 21 by § 112. But 21 = 22 by hyp. :.23 = 24 by Ax. 11.]

20 The bisectors of vertical angles form one straight line. (Converse of Ex. 19.)

21 The bisectors of supplementary adjacent angles are perpendicular to each other. [< a = Z a', and ≤ b = Zb' by hyp. To prove a + ≤ b = a rt. Z. What is the

sum of the four ?

a and b are half this sum by hyp.]

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22 The bisectors of complementary adjacent angles include an angle of 45°. [Follow the line of proof given in Ex. 21.]

113 From a point without a line, only one perpendicular can be drawn to the line.

HYPOTHESIS. P is a point without the line ED, PK is perpendicular to ED, and PK' is any other line from P to ED.

Let PK, remaining constantly perpendicular to ED and in the same plane, move on from P until the point K falls on the point K', PK taking the position of P'K'.

"A geometric figure can be moved from one position to another without changing its form, size, or the relation of its parts." Post. 3

Then P'K' is to ED by construction.

.. PK' is not to ED.

"Only one perpendicular can be drawn to a given line at a given point

114 DEFINITION. Parallel lines are lines in the same plane which cannot meet however far produced.

PK'K is an acute

Exercise 23. In the figure above, show that the angle (§ 74), and that the PK'D is an obtuse angle (§ 75).

PROPOSITION V. THEOREM

115 Two straight lines in the same plane, perpendicular to the same straight line, are parallel.

HYPOTHESIS. a and b are two lines each perpendicular to the line m.
CONCLUSION. a and b are parallel.

PROOF

Could a and b, upon being produced, meet at some point, as at X, there would be two perpendiculars drawn from the point X to the line m, which is impossible.

“From a point without a line only one perpendicular can be drawn to the

line."

... a and b cannot meet.

... a and b are parallel.

§ 113

"Parallel lines are lines in the same plane which cannot meet however

116 SCHOLIUM. Proposition V is proved by the indirect method, called "reductio ad absurdum."

The truth of the theorem is established by proving that the supposition that the theorem is not true leads to an absurdity. Thus, we suppose that the lines a and b are not parallel, i.e. that they will, if sufficiently produced, meet. The supposition leads to the conclusion that two perpendiculars can be drawn from the same point to the same straight line. The conclusion is false; therefore the supposition is false and must be abandoned.

The logic is this: If it is not true that the lines are not parallel, then it is true that the lines are parallel.

PROPOSITION VI. THEOREM

117 A straight line perpendicular to one of two parallels is perpendicular to the other.

HYPOTHESIS. AB and CD are parallel lines, and GH is perpendicular to AB. CONCLUSION. GH is perpendicular to CD.

PROOF

Through H draw EFL to GH.

"A straight line can be drawn from any point in any direction to any

extent."

Then EF is I to AB.

Post. 2

"Two straight lines in the same plane perpendicular to the same straight line are parallel."

But CD is to AB by hyp.

.. EF and CD coincide, for

each passes through the same point H.

§ 115

"Through the same point only one straight line can be drawn parallel to

PROPOSITION VII. THEOREM

118 Two straight lines parallel to a third straight line are parallel to each other.

119 A transversal is a straight line which intersects two or more straight L lines. Thus, the line T is a transversal to the lines L and L', and the eight angles formed at the points of intersec- L tion are named as follows:

8/h

Post. 2

§ 117

$ 115

Q. E. D.

a/b

c/d

T

120 Interior angles are those within the two lines; as, angles c, d, e, and f. Exterior angles are those without the two lines; as, angles a, b, g, and h. Alternate interior angles are those within the two lines and on opposite sides of the transversal, but not adjacent; as, angles c and f, d and e. Alternate exterior angles are those without the two lines and on opposite sides of the transversal, but not adjacent; as, angles a and h, b and g. Corresponding angles are those similarly situated with respect to the two lines and on the same side of the transversal; as, angles a and e, c and g, b and ƒ, d and h.