Proposition 6. Theorem. 60. Two oblique lines drawn from a point to a straight line, cutting off equal distances from the foot of the perpendicular, are equal. Hyp. Let CD be the from C to the line EF, and CA and CB two oblique lines so that AD = DB. Proof. Let the part CAD be revolved about CD as an axis until it comes into the plane of CDB. 61. COR. Two equal oblique lines drawn from a point to a straight line, make equal angles with that line, and cut off equal distances from the foot of the perpendicular, Proposition 7. Theorem. 62. If from any point two lines be drawn to the ends of a straight line, their sum will be greater than the sum of two other lines similarly drawn, but included by them. A B Hyp. Let AB, AC be two lines drawn from the point A to the ends of the line BC, and DB, DC two lines similarly drawn, but included by AB, AC. To prove AB+ AC > DB + DC. Proof. Produce BD to meet AC at E. 1. Write out in full the proof of the second part of Prop. 4, that the angle AEC is equal to the angle BED. 2. Prove that the straight line which bisects the angle AEC, in Prop. 4, bisects also the vertical angle BED, Proposition 8. Theorem. 63. Of two oblique lines drawn from the same point to the same straight line, that which meets the line at the greater distance from the foot of the perpendicular is the greater. P Proof. Produce PC to P', making CP' = PC. Because AB is to PP' at its middle point, But ... PD = DP', and PE EP'. (60) (62) PEEP' > PD + DP'. ... 2PE > 2PD, ... PE> PD. If the two oblique lines are on opposite sides of PC, as PE and PD', and if CE > CD', take CD = CD', and join PD. Then But 64. COR. 1. PD = PD'. PE > PD, as just proved. ... PE> PD'. (60) Q. E.D. Only two equal straight lines can be drawn from a point to a straight line. 65. COR. 2. Of two unequal oblique lines, the greater cuts off the greater distance from the foot of the perpendicular. Proposition 9. Theorem. 66. 1. Every point in the perpendicular erected at tre middle of a straight line is equally distant from the extremities of the line. 2. Every point without the perpendicular is unequally distant from the extremities of the line. Hyp. Let DC be 1 to AB at its middle point C, P any point in DC, and O any point without DC. Draw PA, PB, and OA, OB. (1) To prove E Proof. Because DC 1 to AB, and AC = CB, Proof. OA will cut the 1 DC at E; join EB. (60) (Ax. 10) (60) Q. E.D. 67. COR. Every point equally distant from the extremities of a straight line lies in the perpendicular bisector of the line. If a straight line have two points, each of which is equally distant from the extremities of a second line, it will be perpendicular to the second line at its middle point. EXERCISES. 1. Prove that the bisectors of two vertical angles are in the same straight line. 2. Prove that the bisectors of two supplementary adjacent angles are perpendicular to each other. Prove that the internal and external bisectors of an angle are at right angles to each other. 3. Show that the angles AOE and BOD are complementary. 4. Show that the angles AOD and COD are supplementary; and also that the angles BOE and COE are supplementary. 5. If the angle BOD is 37°, how many degrees are there in AOE? 6. If two angles are supplementary, and the greater is 9 times the less, how many degrees are there in each angle? 7. If an angle is 11 times its complement, how many degrees does it contain? PARALLEL LINES. 68. Parallel straight lines are such as lie in the same plane, and never meet, however far they are produced in both directions. 69. A straight line crossing several other lines is called a transversal; as EF. When two straight lines are cut by a transversal, eight angles are formed, which are named as follows: The four angles a, b, g, h, without the two lines, are called exterior angles. a/b A d/c D e The four angles c, d, e, f, within the two lines, are called interior angles. The pair c and e, and the pair d and f, are called alternate-interior angles. The pair a and g, and the pair b and h, are called alternate-exterior angles. The pairs a and e, b and ƒ, c and g, d and h, are called corresponding angles.* *Called also exterior-interior angles. |