MATHEMATICS. PLANE GEOMETRY. [**>* i. PROP, xvn.- 



trr.ix - CEFrf . 



\ 



A B C F,J ami the angle A 

 Pr. . ECF: butACD 

 ii greater than EOF, .'. 

 A C 1) u greater than A ; 

 . . if one tide, B C, of a 

 triangle be prolonged, the 

 exterior angle A C D U 

 greater than the interior 

 . o angle A oppotilt to the tick 

 prolonged : but if the aide 

 prolonged be A C, the ei- 

 terior angle will be B C O, 

 and the interior angle op- 

 posite, ABC;.-.BCGis 

 greater than ABC. But 

 Pr.is. ACD-BCG,*.-. A CD U greater than ABC; 

 and it was l>cfure proved that A C D U greater also than 

 A, . . A C D u yrtuter than either of the interior opposite 

 angle* B A C, A b C. Therefore, if one tide, ic. Q.E.D. 



PROPOSITION XVII. THEORBM. 



Any two angle* of a triangle (A B C) are together less than 



two right angle*. 



Prolong one of the sides, as B C to D ; then the ex- 

 A tenor angle A C 1) is greater 



than the interior opposite 



Pr. 16. angle B.* To 

 each of these add A C B ; 

 then A C D, A C B are to- 

 gether greater than B, 



, Ax 4 ACBjfbutACD, 

 A C B are together equal to 



Pr. IS. two right angles ;* . . B, A C B are together 

 ' leu than two right angles. And if B A be prolonged, it 

 may be proved, in like manner, that A, B are together 

 less than two right angles ; or if C A be prolonged, that 

 A and A C B are together leas than two right angles. 

 Therefore, any two atiglet, <tc. Q.E.D. 



PROPOSITION XVIII THEOREM. 



The greater tide of every triangle (A B C) u oppotiU to the 



greater angle. 



Lt A C be greater than A B ; the angle ABC shall I,,. 

 A greater than the angle C. 



From A C the greater 

 cut off A D = A B the less, 

 and draw B D. Then be- 

 cause A D B is an exterior 

 angle of the triangle BDC, 

 ~ 'Pr.ie. it is greater than C:* 

 p r . s. but A I) B = A B D,t .' . the angle A B D is 

 likewise greater than C ; much more, then, is the angle 

 A B C greater than C. Therefore, the greater tide, <tc 

 QE.D. 



PROPOSITION XIX. THEOREM. 



The greater angle of every triangle (ABC) it subtended 

 by the greater tide, or has the greater side opposite to it. 

 Let the angle B be greater than C ; then A C shall be 

 greater than AB. 



For, if it be not greater, 

 A C must either be equal to 

 A B, or less than it. If it 

 were eirual, the angle B 



Pr. S. would = C ;* but 

 it is not, .-.AC u not = 

 AH. If it were kit, the 



Pr. u. C rf but it is not, 



. , 



angle B would be less than 

 ..... ... ,, . .. ... ..., . . A C w not let* than A B ; 



and it was shown that it is not equal to A B, . . A C u 

 greater than AB, . . the greater angle, <tc. Q.E.D. 



PROPOSITION XX. THEOREM. 



Any two tide* (B A, A C)o/ a triangle (A B C) ore together 



greater than the third tide (B C). 

 Prolong one of the two sides, as B A, to D, and make 



Pr. S. AD AC * the other of the two sides : join 

 D.C. 



Because A D - A C, the angle 

 4 Pr. . A CD D ; t but tlie 

 angle BCD is greater than A CD, 

 . ' . B C D is greater than D. And 

 because the angle BCD of the 

 triangle D B C is UT -ater than the 

 angle D, and th.it the greater 

 angle is subtended by the greater 

 Pr. i side,* the side BD is 

 greater than B C : but B D - B A + A D - B A + A C, 

 . ' . B A + A C it greater than B C. Therefore, any two 

 sides, <tc. Q.KD. 



PROPOSITION XXI. THEOREM. 



If from the endt of a tide of a triangle there be drawn two 

 straight line* (B D, C D) to a [mint within the tnanijlr, 

 these shall be together lest than the other two tides (AB, 

 AC) of the triangle, but shall contain a greater angle. 

 Prolong B D to E. The two sides B A, A E of the 

 triangle ABE are together greater 

 Pr. 20. than BE.* To each of 

 these unequals add E C, . . B A, AC 

 are greater than BE, EC. Again : 

 the two sides C E, E D of the tri- 

 angle C E D are together gr. 



Pr. Jo. than C D.f To each of 



B - c those add DB, .-. C E, E B are 



greater than C D, D B . But it was 

 shown that B A, A C are greater than B E, E C ; much 

 more then are B A, A C greater than B D, DC. Again : 

 the exterior angle B D C of the triangle C D E is greater 



Pr. 16. than C E D ;* and the exterior angle C E B 

 of the triangle A B E is greater than B A C :* much more 

 then is B D C greater than B A C. Therefore, if from 

 the ends, ic. Q 1 . 1 ). 



PROPOSITION XXII. PROBLEM. 

 To make a triangle of which the sides shall be equal to 

 three given straight lines (A, B, C), each to each, but any 



Pr. so. two of tliese mutt be greater Umn the tlvird* 

 Take a straight line D E, terminated at the point D, 



but unlimited towards 

 E, and make 1) F = 

 A, F0= It, and (i II 

 i-r. s. C.t With 

 c.-ntre F and radius 

 F 1 1 describe the circle 

 DKL ; and with centre 

 G and radius G II de- 

 scribe the circle H KL, 

 cutting the former in K. Draw K F, K G : the triangle 

 K. I-' U has its three sides equal to the three lines A, B, C. 

 Because F is the centre of the circle DKL, F D = 

 F K ; but F D A, . . F K = A. Again : because G is 

 the centre of the circle L K H, G H = G K ; but G H = 

 C, . . G K = C ; but F G = B, . . the sides of the tri- 

 angle K F G ore equal to A, B, C, each to each. Which 

 was to be done. 



NUIK. U U plain that the circles could not cut unless their radii were 

 together greater than the distance between their centre* : if huch 

 were not the case, they would be wholly without one another. And 

 if the distance of the centres, together with one radius, were smaller 

 thiiu or equal to the other radius, one circle would be wholly wit AIM 

 the other : the hypothesis precludes both of these circumstances. 



PROPOSITION XXIII. PROBLEM. 

 At a given jxtiiit (A) in a given straight line (AB) to make 



an angle equal to a given angle (C). 

 In C D, C E take any points D, E, one in each, and 

 draw D E. Make the tri- 

 A angle A F G, the sides of 



which shall be equal to 

 those of C D E namely, 

 AF-CD, AG = CE, 

 Pr.22. andFG = DE;* 

 then the angle A shall be 

 equal to C. 



I :, muse AF-CD. A G 



CK, 



Pr. 8. the angle A = the angle C,* 

 point, &c. It'/i.e'i wot to be done. 



at the given 



