782 



THE IRRIGATION AGE. 



3. Parallels and Quadrilaterals. 



A figure bounded by four intersecting straight lines is 

 called a Quadrilateral ; thus, ABCD, in Fig. 15, is 

 a quadrilateral ; it has four sides and four angles ; 

 a line like AC is called a diagonal; this diagonal 

 cuts the quadrilateral into two triangles which 

 shows that the sum of the four angles of any 

 quadrilateral equals 360 (two triangles). 

 Fig. is. If two lines, AB and CD, are cut by a third 



line EF, so that the sum of the inner angles a + b 180", 

 then the lines AB and! CD are called parallel, which means 



/they will not intersect each other no matter 

 j how far they are extended. It is easily seen 

 t] ~ that angle a = angle c, and angle b = angle d 

 / as well when AB and CD are parallel. 



f -jrf- -// If in a quadrilateral the opposite sides 



> are parallel it is called a parallelogram; thus, 



Fig. 16. if in Fig. 17, AB is parallel to DC and AD is 



parallel to BC, then ABCD is a parallelogram and the oppo- 



j site sides are equal ; also, the opposite 



/ angles are equal ; hence, AB = DC and AD 



/ / := BC; also, angle a = angle c and angle 



j, c b = angle d; furthermore, if the two 



Fig. IT. diagonals are drawn, AC and BD (See Fig. 



18), intersecting in O, then AO OC and BO = 

 DO, which means the diagonals bisect each other. 

 If each angle in a quadrilateral equals 90, 

 then it is called a rectangle or oblong (See 

 Fig. 19.) All the properties of the parallelo- 

 gram belong to this figure as well, and it has 

 one special feature which is, that the diagonals 

 Fig. 18. i n a rectangle are equal. Thus, AC = BD. 

 This principle is made use of to test rooms or other rec- 

 tangular areas to see whether they are square; 

 for instance, let ABCD represent a room and 

 let it be tested if it is square; take a string 

 from the corner A to corner C, then take 

 . same string and stretch it from B to D; 

 the string will fit both ways if the figure is 

 a rectangle; if not, proves that the corners are not right 

 angles. If in a rectangle all the sides are equal, it is called 

 a square ; thus, if in Fig. 20, AB = BC CD == AD, then 

 _g the figure is a square ; all the properties of the 

 rectangle apply also to the square with this addi- 

 tional one that the diagonals AC and BD intersect 

 each other under right angles, and that the angles 

 formed by the diagonals with the sides of the 

 square are all 45. 



If in a parallelogram all the sides are equal it 

 it called a Rhombus. Thus, if in Fig. 21, AB = BC 

 = CD = AD, the figure is a Rhombus, and all the 

 properties of the parallelogram apply to this figure 

 with this one addition, that the two diagonals 

 AC and BD stand perpendicular to each other. 



If in a quadrilateral two sides are parallel, 

 but the other two are not, like in Fig. 22, 

 it is called a trapesoid, if AB and DC 

 are parallel and the middle point M of 

 AD is connected with the middle point 

 N of BC, then the line MN is parallel 

 AB + CD 



to AB, and CD and MN = 



2 

 4. Mensuration of Polygons. 



Any geometrical figure bounded by straight lines is called 

 a Polygon. Any such closed figure embraces a certain part of 

 the plain surface called area. Areas are measured by some 

 unit area, which is the square whose side is the unit of 

 length ; thus, a square whose side is an inch, is called a 

 square inch; the square whose side is 1 foot, is called a 

 square foot ; the square whose side is 1 meter long, is called 

 1 square meter, etc. If thus, one polygon is found to contain 

 8 square inches and another contains 16 square inches, then 

 the former is half the size of the latter. 



It is evident, that in a rectangle 2 inches wide and 4 

 inches high (See Fig. 23), the area will be 8 

 square inches, because by drawing the lines EF, 

 GH, JK and LM from the inch points parallel with 

 the sides of the rectangle the figure is cut into 8 

 equal squares, each equal to 1 square inch; hence, 

 the area of a rectangle is found by multiplying 

 Fig. 23. the base AB, by the height BC. The area of a 



Fig. 20. 



Fig. 22. 



Fig. 24. 



parallelogram is found also by multiplying the base by the 

 height, but care must be taken to take the 

 height correctly. Thus, in Fig. 24, if AB is 

 taken for the base, then EF is the height, which 

 is the perpendicular distance between AB and 

 CD; if AD is taken as the base, then GH must 

 be taken as the height. 



Any parallelogram can be divided into two 

 equal triangles, as in Fig. 25, where the 

 line AC divides the parallelogram ABCD 

 into two equal triangles ABC and ADC, 

 so each triangle is equal to half of 

 the area of the parallelogram. Let CE 

 be the perpendicular distance between AD and 

 BC, then the area of the triangle ACD = 

 ADXCE 

 Flg 25. ' which gives the rule that the area 



of a triangle is found by multiplying base by height and 

 divide by 2. Let in Fig. 26, the length of the three sides be 

 a, b and c respectively, then each of them can be considered 

 as base, and the right height is the per- 

 pendicular drawn to such base from the 

 opposing vertex ; if then three heights are 

 drawn they pass through the same point O; 

 let h be the height from A to a, h' the 

 height from point B to b and h" the 

 height from point C to c, then the area of 

 Fis- 26. a h bh 



the triangle ABC can be expressed as follows : and 



2 2 

 ch" 



, all of which give the same area and are therefore equal ; 

 2 



ah bit' ch 



hence : = = . 



222 



Any polygon can be divided into triangles. In Fig. 27 is 

 shown a heptagon (seven-sided polygon), 

 ABCDEFG; by drawing the diagonals AC, 

 AD, AE, AF, the polygon is cut into five 

 triangles and by finding the area of them and 

 adding them gives the area of the heptagon. 

 Let AC = a, AD = b, AE c, and AF = 

 d; also, the height from B to a = h, to b 

 from C = h', to c = h" from D and F to 

 from G to d ;= /!*, then the area A of the 

 heptagon would be expressed as follows : 

 ah ch' ch" ch'" dh* 



A I I I I 



" i \ V I 



22222 

 In this expression the factor '/ can be set out: 

 A ~ % (ah + bh' + ch" +'ch"' + rf/i 1 ). 

 This principle can be used for any polygon. 



5. Proportionality. 



Things of the same kind can be compared with each other 

 numerically; thus, if a line AB, in Fig. 28, is 4" 

 long, and a line CD, is 8" long, then CD is, twice 

 as long as AB, or AB is y* of CD; this is called 

 a ratio. Now, let EF be 3" and GH be 6" long, then 

 the ratio of EF to GH is also the same as that of 

 AB to CD; hence, the four lines shown in Fig. 28 

 form two equal ratios and can be combined into 

 the proportion : 

 AB :CD=EF: GH. 

 (Read: AB is to CD as EF is to GH}. 

 The four quantities are called the terms of the propor- 

 tion; AB is the First Antecedent, CD the First Consequent, 

 EF the Second Antecedent and GH the Second Consequent. 

 In reality a proportion consists of two equal fractions. As 

 above AB : CD = 4/8, and EF : GH = 3/6, or 4/8 = 3/6. 

 The principal rule relating to proportions is that the 

 product of the two extreme members is equal to the product 

 of the middle (mean) members; so, if 3 : 6 = 4 : 8, then 

 3X8 = 6X4. 



This rule is used to find an unknown member in a pro- 

 portion ; for instance, in Fig. 28 let the length of the line CD 

 be unknown and let it = x; then 



3 : A- = 4 : 8 



form the product of the mean and extreme members; 



4 x = 24; 



divide both sides by 4 gives x = 6. 



(Continued on Page 798.) 



2 



Fig. 28. 



