GEOMETRY. 



405 



sy 



B D 



In the two triangles ABC, DEF, fig. 58, AB : 

 DE : : AC : DF : : BC : EF ; the two triangles will 

 have their corresponding angles equal. 



Properties of quadrilaterals, 8fC, Let ABCD, fig. 

 59, be a parallelogram, of which the diagonal isBD; 

 then will its opposite sides and angles be equal to 

 each other, and the diagonal BD wfll divide it into 

 two equal parts, or triangles. If one angle of a par- 

 allelogram be a right angle, all the other three will 

 also be right angles, and the parallelogram a rec- 

 tangle. The sum of any two adjacent angles of a paral- 

 lelogram is equal to two right angles. Let ABCD 

 be a quadrangle, having the opposite sides equal, 

 namely, the side AB equal to DC, and AD equal to 

 BC: then shall these equal sides be also parallel, 

 and the figure a parallelogram. Let AB, DC, be 

 two equal and parallel lines ; then will the lines AD, 

 BC, which join their extremities, be also equal and 

 parallel. Let ABCD, ABEF, fig. 60, be two paralle- 

 lograms, and ABC, ABF, two triangles, standing on 

 the same base AB, and between the same parallels 

 AB, DE ; then will the parallelogram ABCD be 

 equal to the parallelogram ABEF, and the triangle 

 ABC equal to the triangle ABF. Parallelograms, 

 or triangles, having the same base and altitude, are 

 equal. Parallelograms, or triangles, having equal 

 bases and altitudes are equal. 



63 



64 



Aii ABEF AH B 



Let ABCD, fig. 61, be a parallelogram, and ABE 

 a triangle, on the same base AB, and between the 

 same parallels AB, DE ; then will the parallelogram 

 ABCD be double the triangle ABE, or the triangle 

 half the parallelogram. A triangle is equal to half 

 a parallelogram of the same base and altitude, be- 

 cause the altitude is the perpendicular distance be- 

 tween the parallels, which is everywhere equal, by 

 the definition of parallels. If the base of a paralle- 

 logram be half that of a triangle, of the same alti- 

 tude, or the base of a triangle be double that of the 

 parallelogram, the two figures will be equal to each 

 other. Let AC, EG, fig. 62, be two rectangles, 

 having the sides AB, BC, equal to the sides EF, 

 FG, each to each ; then will the rectangle AC be 

 equal to the rectangle EG. Let AC, fig. G3, be a 

 parallelogram, BD a diagonal, EIF parallel to AB 

 or DC, and GIH parallel to AD or BC, making AI, 

 1C, complements to the parallelograms EG, HF, 

 which are about the diagonal DB: then will the 

 complement Al be equal to the complement 1C. 

 Let ABCD, fig. 64, be a parallelogram, whose dia- 

 gonals intersect each other in E : then will AE be 

 equal to EC, and BE to ED ; and the sum of the 

 squares of AC, BD, will be equal to the sum of the 

 squares of AB, BC, CD, DA. That is, 



AE = EC, and BE = ED, 

 and AC 8 + BD' = AB' -f BC + CD' + DA'. 



A trapezoid is equal in area to the rectangle of its 

 altitude and half the sum of its parallel sides. The 

 side of a square is incommensurate with its diagonal. 



60 



67 



Let ABCD, fig. 65, be a quadrangle ; then the 

 sum of the four inward angles, A + B + C + Dis 

 equal to four right angles. Let ABCDE, fig. 66, 

 be any figure ; then the sum of all its inward angles, 

 A-f-B + C + D + E, is equal to twice as many 

 right angles, wanting four, as the figure has sides. 

 Let A, B, C, &c., fig. 67, be the outward angles of 

 any polygon, made by producing all the sides ; then 

 will the sum A + B + C -f D + E, of all those 

 outward angles, be equal to four right angles. 



68 69 70 



Properties of Circles, &c- Let AB, fig. 68, be 

 any chord in a circle, and CD a line drawn from the 

 centre C to the chord. Then, if the chord be bisected 

 in the point D, CD will be perpendicular to AB. 

 Let ABC, fig. 69, be a circle, and D a point within 

 it ; then if any three lines, DA, DB, DC, drawn 

 from the point D to the circumference, be equal to 

 each other, the point D will be the centre. Let two 

 circles touch one another internally in the point; 

 then will the point and the centres of those circles 

 be all in the same right line. Let two circles touch 

 one another externally at a point; then will the 

 point of contact and the centres of the two circles be 

 all in tho same right line. Let AB, CD, fig. 70, be 

 any two chords at equal distances from the centre 

 G; then will these two chords AB, CD, be equal to 

 each other. 



Let the line ADB, fig. 7.1, be perpendicular to the 

 radius CD of a circle ; then shall AB touch the cir- 

 cle in the point D only, and be a tangent. Let AB, 

 fig. 72, be a tangent to a circle, and CD a chord 

 drawn from the point of contact C ; then is the angle 

 BCD measured by half the arc CFD, and the angle 

 ACD measured by half the arc CGD. Let BAC, 

 fig. 73, be an angle at the circumference; it has for 

 its measure, half the arc BC which subtends it. 



75 



76 



Let C and D, fig. 74, be two angles in the same 

 segment ACDB, or, which is the same thing, stand- 

 ing on the supplemental arc AEB ; then will the 

 nngle C be equal to the angle D. Let C, fig. 75, 

 be an angle at the centre C, and D an angle at the 

 circumference, both standing on the same arc or 

 same chord AB ; then will the angle C be double of 

 theangleD ortheangleDequal to half the angle C. 



