and since that is, <t=2v, 2Zt+Zw= 2 rt. 4; 2t+<w=1 rt. ≤ ; Zt is less than a rt. Z. PD is not perpendicular to AB. But since PD represents any line from P to AB other than PC, PC is the only perpendicular that can be drawn to AB from P. Therefore, etc. Q.E.D. 61. Cor. A perpendicular is the shortest line that can be drawn from a point to a line. 1. Since PCF is a straight line, is PDF a straight line? Ax. 12. 2. Which line, then, is the shorter, PCF or PDF? 3. What part of PCF is PC? Of PDF is PD? 4. Then, how do PC and PD compare in length? Ax. 10. 5. Since PD represents any line from P to AB other than the perpendicular PC, what is the shortest line that can be drawn from a point to a line? 62. The distance from a point to a line is always understood to be the perpendicular or shortest distance. PARALLEL LINES 63. Lines which lie in the same plane, and which cannot meet however far they may be extended, are called Parallel Lines. 64. A straight line which crosses or cuts two or more straight lines is called a Transversal. EF is a transversal of AB and CD. /S t/v E B D Eight angles are formed by the transversal EF with the lines AB and CD. w/x 31/2 C 65. The angles above AB and those below CD, or those without the two lines cut by the trans versal, are called Exterior Angles. Angles r, s, y, and z are exterior angles. 66. The angles between, or within the two lines cut by the transversal, are called Interior Angles. Angles t, v, w, and x are interior angles. 67. Non-adjacent angles without the two lines, and on opposite sides of the transversal, are called Alternate Exterior Angles. Angles r and z, or s and y, are alternate exterior angles. 68. Non-adjacent angles within the two lines, and on opposite sides of the transversal, are called Alternate Interior Angles. Angles t and x, or v and w, are alternate interior angles. 69. Non-adjacent angles, which lie one without and one within the two lines, and on the same side of the transversal, are called Corresponding Angles. Angles r and w, s and x, t and y, or v and z, are corresponding angles. Corresponding angles are also called Exterior Interior Angles. 70. Ax. 13. Through a given point but one straight line can be drawn parallel to a given straight line. Proposition VII 71. Draw a straight line; also two other lines each perpendicular to the first line. In what direction do the perpendiculars extend with reference to each other? Theorem. If two straight lines are perpendicular to the same straight line, they are parallel. Proof. Since, by data, both CD and EF are perpendicular to AB, they cannot meet, for, if they should meet, there would then be two perpendiculars from the same point to the line AB, which is impossible. § 60. Hence, § 63, CD || EF. Q.E.D. Proposition VIII 72. Draw two parallel lines; also a transversal perpendicular to one of them. What is the direction of the transversal with reference to the other parallel line? Theorem. If a straight line is perpendicular to one of two parallel straight lines, it is perpendicular to the other. Proof. If EF is not perpendicular to CD at the point J, it will be perpendicular to some other line drawn through that point. Suppose GH is that line. then, § 70, GH and CD passing through J cannot both be parallel to AB. Hence, the hypothesis that EF is not perpendicular to CD is untenable. Consequently, EF CD. Therefore, etc., Q.E.D. Ex. 18. Two lines are drawn each parallel to AB, and another line making an angle of 90° with AB. What is the direction of this line with reference to each of the other two lines? Ex. 19. State and illustrate the differences between a plumb line, a perpendicular line, and a vertical line. Ex. 20. Two parallel lines are cut by a third line making one interior angle 35o. What is the value of the adjacent interior angle? Proposition IX 73. 1. Draw two parallel lines; also a transversal. How many pairs of vertical angles are formed? How many pairs of supplementary adjacent angles? How many sizes of angles are formed? How many angles of each size? When may they all be of the same size? 2. Name a pair of angles whose sum is equal to two right angles. Name seven other pairs. Name a set of four angles whose sum is equal to four right angles. Name three other sets. 3. Name the pairs of alternate interior angles. How do the angles of any pair compare in size? Theorem. If two parallel straight lines are cut by a transversal, the alternate interior angles are equal. Data: Any two parallel straight lines, as AB and CD, cut by a trans- 4versal, as EF, in the points H and J. To prove the alternate interior an gles, as AHJ and DJH, equal. E G H B Proof. Through L, the middle point of HJ, draw GK 1 CD. Revolve the figure JLK about the point I and apply it to the figure HLG, so that LJ coincides with LH. Then, since, § 59, JLK = ZHLG, LK takes the direction of LG. Const., JK GK and HG GK, and since the point J falls upon the point H, 74. If two theorems are related in such a way that the data and conclusion of one become the conclusion and data, respectively, of the other, the one is said to be the converse of the other. Thus, the converse of the theorem just proved is, "Two straight lines cut by a transversal are parallel, if the alternate interior angles are equal." Converse propositions cannot be assumed to be true. They may be true, but their truth must be established by proof. Thus, the truth of the proposition, "The product of two even numbers is an even number," can be established readily, but its converse, "An even number is the product of two even numbers," is evidently false. Proposition X 75. Draw two lines; also a transversal. In what direction do the lines extend with reference to each other, if the alternate interior angles are equal? Theorem. Two straight lines cut by a transversal are parallel, if the alternate interior angles are equal. (Converse of Prop. IX.) Data: Two straight lines, as AB and CD, such that when cut by any trans- Aversal, as EF, the alternate interior Kangles, as AHF and EJD, are equal. To prove AB and CD parallel. E -B H Proof. If AB is not parallel to CD, then some other line, as KL, drawn through the point H is parallel to CD. which is absurd, since a part cannot be equal to the whole. Hence, the hypothesis, that some other line, as KL, drawn through the point H is parallel to CD, is untenable. Consequently, Therefore, etc. AB | CD. Q.E.D. |