Perpendicularity is denoted by the symbol L, to be read "perpendicular to" or "is perpendicular to." A right angle is denoted by the symbol. The symbol also denotes two right angles or a straight angle. Def. 3.-When two angles together make up a right angle they are complementary to one another, and each is the complement of the other. The right angle is the simplest of all angles, for when two lines form an angle they form four angles equal in opposite pairs. But if any one of these is a right angle, all four are right angles. Perpendicularity is the most important directional relation in the applications of Geometry. Def. 4.-An acute angle is less than a right angle, and an obtuse angle is greater than a right angle, and less than two right angles. 41°. From (36°) we have I circumangle = 2 straight angles =4 right angles. In estimating an angle numerically it may be expressed in any one of the given units. If a right angle be taken as the unit, a circumangle is expressed by 4, i.e. four right angles, and a straight angle by 2. Angles less than a right angle may be expressed, approximately at least, by fractions, or as fractional parts of the right angle. For practical purposes the right angle is divided into 90 equal parts called degrees; each degree is divided into 60 equal parts called minutes; and each minute into 60 equal parts called seconds. Thus an angle which is one-seventh of a circumangle contains fifty-one degrees, twenty-five minutes, and forty-two seconds and six-sevenths of a second. This is denoted as 42°. Theorem.-Through a given point in a line only one perpendicular can be drawn to the line. The line OC is LAB, and OD is any other line through O. C D A 10 B Then OD is not AB. Proof-The angles BOC and COA are each right angles (40°, Def. 2). Therefore BOD is not a right angle, and OD is not ↓ AB. q.e.d. Def. The perpendicular to a line-segment through its middle point is the right bisector of the segment. Since a segment has but one middle point (30°, Ex. 3), and since but one perpendicular can be drawn to the segment through that point, .. a line-segment has but one right bisector. 43°. Def.-The lines which pass through the vertex of an angle and make equal angles with the arms, are the bisectors of the angle. The one which lies within the angle is the internal bisector, and the one lying without is the external B E * F D bisector. Let AOC be a given angle; and let EOF be so drawn that LAOE =LEOC. A EF is the internal bisector of the angle AOC. Also, let GOH be so drawn that LCOG=4HOA. HG is the external bisector of the angle AOC. and LCOG=LHOA _HOA=2GOB, 4COG=4GOB; (hyp.) (40°) and the external bisector of AOC is the internal bisector of its supplementary angle, COB, and vice versa. The reason for calling GH a bisector of the angle AOC is given in the definition, viz., GH makes equal angles with the arms. Also, OA and OC are only parts of indefinite lines, whose angle of intersection may be taken as the LAOC or as the COB. 44. Just as in 23° we found two points which are said to divide the segment in the same manner, so we may find two lines dividing a given angle in the same manner, one dividing it internally, and the other externally. Thus, if OE is so drawn that the LAOE is double the LEOC, some line OG may also be drawn so that the LAOG is double the LGOC. This double relation in the division of a segment or an angle is of the highest importance in Geometry. 45°. Theorem.-The bisectors of an angle are perpendicular to one another. EF and GH are bisectors of the LAOC; 1. Three lines pass through a common point and divide the plane into 6 equal angles. Express the value of each angle in right angles, and in degrees. 2. OA and OB make an angle of 30°, how many degrees are there in the angle made by OA and the external bisector of the angle AOB? 3. What is the supplement of 13° 27′ 42′′? What is its complement? 4. Two lines make an angle a with one another, and the bisectors of the angle are drawn, and again the bisectors of the angle between these bisectors. What are the angles between these latter lines and the original ones? 5. The lines L, M intersect at O, and through O, L' and M' are drawn respectively to L and M. The angle between L' and M' is equal to that between L and M. SECTION III. THREE OR MORE POINTS AND LINES. 46°. Theorem.—Three points determine at most three lines ; and three lines determine at most three points. Proof 1.-Since (24°, Cor. 2) two points determine one line, three points determine as many lines A M N B as we can form groups from three points taken two and two. Let A, B, C be the points; the groups are AB, BC, and CA. Therefore three points determine at most three lines. 2.—Since (24°, Cor. 3) two lines determine one point, three lines determine as many points as we can form groups from three lines taken two and two. But if L, M, N be the lines the groups are LM, MN, and NL. Therefore three lines determine by their intersections at most three points. 47°. Theorem.--Four points determine at most six lines; and four lines determine at most six points. Proof.-1. Let A, B, C, D be the four points. The groups of two are AB, AC, AD, BC, BD, and CD; or six in all. Therefore six lines at most are determined. 2. Let L, M, N, K be the lines. The groups of two that can be made are KL, KM, KN, LM, LN, and MN; or six in all. Therefore six points of intersection at most are determined. A K D N C M Cor. In the first case the six lines determined pass by threes through the four points. And in the second case the six points determined lie by threes upon the four lines. This reciprocality of property is very suggestive, and in the higher Geometry is of special importance. Ex. Show that 5 points determine at most 10 lines, and 5 lines determine at most 10 points. And that in the first case the lines pass by fours through each point; and in the latter, the points lie by fours on each line. 48°. Def.-A triangle is the figure formed by three lines. and the determined points, or by three points and the determined lines. The points are the vertices of the triangle, and the linesegments which have the points as end-points are the sides. The remaining portions of the determined lines are usually spoken of as the "sides produced." But in many cases generality requires us to extend the term "side" to the whole line. B F extending outwards as far as required, are the sides produced. |