Plane.' Planus, level. another. SECTION I. THEOREM I. Part. En. Let ABC, PRS be right angles ; it is required to prove that ABC is equal to PRS. Proof. If the point B were placed on the point R, and the line BC along the line RS, then because the lines DC, QS are 'straight, (Ax. 2.) therefore the angle DBC coincides with, and is equal to, the angle QRS. (Ax. 1.) But by Def. 14, the angle ABC is half the angle DBC; and the angle PRS is half the angle QRS; and the halves of equals are equal ; therefore the right angle ABC is equal to the right angle PRS. Q. E. D. CoR. I. At a given point in a given straight line there can be only one perpendicular drawn to that line. Cor. 2. The complements of equal angles are equal. THEOREM 2. If a straight line stands upon another straight line it makes the adjacent angles together equal to two right angles*. Part. En. Let DB stand upon the straight line AC; it is required to prove that the adjacent angles ABD, DBC are together equal to two right angles. À o Proof. Because ABC is a straight line, (Hyp.) therefore the angle ABC is equal to two right angles ; (Def. 14.) but the angle ABC is, from the figure, made up of the angles ABD and DBC; (Def. 12.) therefore the angles ABD and DBC are together equal to two right angles. Q. E. D. * Eucl. 1. 13. 20 THEOREM 3. If the adjacent angles made by one straight line with two others are together equal to two right angles, these two straight lines are in one straight line*. Part. En. Let the adjacent angles DBA, DBC made by BD with the two straight lines BA, BC be together equal to two right a angles; it is required to prove that AB, BC are in one straight line. Proof. Because DBA and DBC are together equal to two right angles; (Hyp.) and DBA and DBC, from the figure, make up the angle ABC; therefore ABC is an angle of two right angles; and therefore ABC is a straight line. (Def. 14.) Q. E. D. THEOREM 4. If two straight lines cut one another the vertically opposite angles will be equal to one another. Part. En. Let the straight lines AOB, DOC cut one another, and let AOD, BOC be vertically opposite angles ; it is required to prove that the angle D AOD is equal to the angle BOC. Proof. Because AOB is a straight line ; (Hyp.) therefore the angles AOC and COB are together equal to two right angles. (Th. 2.) * Eucl. 1. 14. And again because DOC is a straight line; (Hyp.) therefore the angles AOC and AOD are together equal to two right angles. (Th. 2.) Therefore the angles AOC and COB are equal to the angles AOC and AOD. Take away the common angle AOC; therefore the angle COB is equal to the angle AOD*. Q. E. D. CoR. The sum of all the angles made by any number of lines taken consecutively which meet at a point is four right anglest. EXERCISES ON ANGLES. 1. If two straight lines intersect at a point, and one of the four angles is a right angle, prove that the other three are right angles. 2. Two straight lines meet at a point. Are the angles at that point together equal to four right angles ? 3. If the four angles made by four straight lines which meet at a point are all right angles, prove that the four lines form two straight lines. 4. If five lines meet at a point and make equal angles with one another all round that point, each of the angles is four-fifths of a right angle. 5. Of two supplementary angles the greater is double of the less; find what fraction the less is of four right angles. 6. Twelve lines meet at a point so as to form a regular twelve-rayed star: find the angle between consecutive rays. * Eucl. I. 15. + Eucl. 1. 15. Cor. 7. If four straight lines 0A, OB, OC, OD meet at a point, and AOB = COD, and BOC=DOA, prove that AOC, BOD are straight lines. 8. Prove that the bisectors of adjacent supplementary angles are at right angles to one another. 9. Find the angle between the bisectors of adjacent complementary angles. 10. Prove that the bisectors of the four angles which one straight line makes with another form two straight lines at right angles to one another. 11. If four lines AO, BO, CO, DO meet at a point 0, and the angles AOB, COD are given equal, and also AO, CO are given as being in the same straight line; prove that BO and DO, if on opposite sides of AOC, are also in the same straight line. 12. If the corner of the page of a book be folded down so as to form an oblique crease, prove that the bisector of the angle between the parts of the edge that meet at the crease will be at right angles to the crease. |