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E. 1 | Hyp. At · B in AB, let BC and
BD make the adjacent
= two rt. angles;
one and the same st.
line CD C. 1 by Sup. Should BD not be it the
same st. line with CB, Pst. 2. make BE in the same st. line with CB. D. 1 by Hyp. ::: AB with the st. line CBE makes the Zs ABC, ABE,
Нур. . But Zs ABC and ABD also = two rt. angles,
Sub. taking away the common ABC, 6 Ax. 3. ... the rem. Z ABE= the rem. _ ABD, 7 ex. abs. and the less Z ABE the greater ABD; which is
impossible; 8 Conc. .. BE is not in the same st. line with CB. 9 Sim. In the same way no line except BD is in the same st. line
with CB; 10 Conc. .:. BD is in one and the same st. line with CB. 11 Rec. Wherefore if at a point in a line, &c.
Q.E.D. Sch.-The words “upon the opposite sides of it" are of essential importance ; for two lines may make with a third, two angles equal to two rt. angles, and yet the two lines not be in one st. line.
3 4 5
PROP. 15.-THEOR. If two st. lines cut one another the opposite, or verticalangles shall be equal.
DEM.–P: 13, Ax. 1, Ax, 3, Ax, 2. Е. І, Нур. . Let the two st. lines AB,
CD intersect in the · E; 2 Conc. then _ AEC = _ DEB, & L CEB = L AED.
LS CEA and AED,
together=twort.angles. 3 Нур. . And :: DE makes with AB the Zs AED, DEB; 4 P. 13. . , these Zs AED and DEB together = two rt. angles;
Ax. 1. also Zs CEA, AED = US AED, DEB:
Sub. Take away the common angle AED, 7 Ax. 3. and rem. Z CEA will = rem. Z DEB.
Sim. In like manner we prove Z CEB = L'AED. 91 Rec. Therefore, if two st. lines cut one another, &c. Q.E.D.
Cor. 1.—The angles formed by two st. lines, AB and CD, cutting each other in one point E, are together equal to four right angles.
D. 11 by P. 13. Zs AEC and CEB together = two rt. angles,
2 Pr, 13. and Zs AED and DEB also two rt. angles;
Cor. 2.- And all the angles formed by any number of st. lines, AC, BC,
given lines, as AC to G,
G .. the Zs ACB, BCG, ACH and HCG toge
ther = four rt. Zs. 2 P. 13. But _s ACB, BCD, DCG, together = US F
E ACB and BCG;
HI 31 P. 13. also Zs ACF, FCE, and ECG, together = LS ACH and
four rt. angles. Sch.-1. This proposition might be briefly proved by saying that the opposite angles are equal, because they have the same supplement.
2. “ This Prop. is the developement of the definition of an angle. If the lines at the angular point be produced, the produced lines have the same inclination to one another as the original lines have.”-Potts' Euclid, p. 49.
3. The converse of this Prop. is, “ If foúr lines meeting in a point make the vertical angles equal, each alternate pair of lines shall be in one and the same st. line."
APP.-To find the breadth of a lake, or the distance between two inaccessible objects, A and B.
If one side of a triangle be produced, the exterior angle is greater than either of the interior opposite ungles.
Con.-P. 10, P. 3, Pst. 1, Pst. 2.-DEM.-P. 15, P. 4, As. 9.
E. 11 Нур.
then the ext. / ACD
and join BE;
and join FC.
and BE = EF, 2 P. 15.
and L AEB= L CEF ; 3 P 4.
..Δ ΑΕΒ A CEF, and L. BAE
But Z ECD or ACD > L ECF;
and .. LACD > _BAC.
Prop. 17.—THEOR. (converse of Ax. 12.)
Con.-Pst. 2, DEM.-P. 16, Ax. 4, P. 13.
Let ABC be a triangle ;
and B, are together < two rt.
angles. C. 1 by Pst. 2. Produce BC to D. D. 1 by Con. :: LACD is the exterior angle,
D 2 P 16... LACD > LB or LA. 3 Add. To each of these unequals add
the L ACB Ax. 4. then Z8 ACD and ACB > Ls ABC and ACB. 5 P. 13. But Zs ACD and ACB two rt. angles.
D. 4. .. Zs ABC, ACB < two rt. angles. 7 Sim. In like manner Zs BAC, ACB < two rt, angles; 8 Sim. and ZS BAC, ABC < two rt. angles. 91
Rec. Therefore, any two angles of a triangle, &c. Q.E.D. Sca. Both the 16th and the 17th propositions will be included in the thirty-second.
Con.-P. 3, ?st. 1.-DEM.-P. 5, P. 16.
In A ABC, let AC be greater
than AB; 2 Conc. then _ ABC is > L ACB. C. 1 byP.3&Pst. 1 AC make AD =
AB, and join BD. D. 1 by C. & P. 5. In A ABD, ::: AD
1... Z ADB
2 P. 16. But ext. ADB > int. _ DCB,
much more is _ ABC > L ACB. 6 Rec. Therefore, the greater side of every triangle, &c. Q.E.D.
ScH.-The argument on which the conclusion depends is named “a fortiori,” by the stronger reason, and proves that the given predicate belongs in a greater degree to one subject than to another; as in the Syllogism,-Y is greater than Z, and X greater than Y; much more is X greater than Z.
PROP. 19.-THEOR. (Converse of the 18th.) The greater angle of every triangle is subtended by the greater side, or has the greater side opposite to it.
DEM.-P. 5. P. 18.
E. 1 Hyp.
In A ABC let ABC be >
SUP.- The side AC must be greater than, equal to, or less than AB. D. 1,by Sup. & P.5. If AC = AB, then _ ABC = L. ACB: 2 Hyp.
but _ABC + L ACB; 3 D. 1 & 2. .. AC + AB. 4 Sup. & P. 18. Again, if AC is < AB, then _ ABC is< L ACB: 5 Hyp.
but ABC is * L ACB; 61 D. 4 & 5. :: AC is * AB. 7 D. 3 & 6. Now, AC is neither equal to, nor less than, AB; 8 Concl.
AC is greater than AB. 9 Rec. Wherefore, the greater angle of every triangle, gc. Q.E.D.
SCH.-1. Props. 5, 6, 18, and 19, may be combined into one thus,—“ One angle of a triangle is greater than, equal to, or less than, another angle, as the side opposed to the one is greater than, equal to, or less than, the side opposed to the other; and vice versa."
2. By Props. 17, 18, and 19, we may, frove that from the same point there can be drawn but one perpendicular to a given line, and that this perpendicular is the shortest of all the lines from the given point to the given line.
3. As from a given point only three lines can be drawn perpendicular to each other, it is impossible to imagine that there are more than three species of quantity,--a Line a Surface, and a Solid.
Con.-Pst. 2, P. 3, Pst. 1.-DEM.P.5, Ax. 9, P. 19.
gether are > BC;
BC and CA > AB.
Pst. 1. join D and C. D. I by C. 1 &P.5.1AD= AC,
.. LADC L ACD;
D. 2 C. 1 & 2 but _ BCD is > LACD ;
3 Ax. 9 .. BCD is > LADC or BDC ; 4 Ax. 9. and in a DBC
Z BCD is > LBDC, 5 P. 19. ... also BD is > BC. 6 C. 1. But BD BA and AC together ; 7 D. 4 & 5. . . BA and AC together are > BC. 8 Sim. In like manner it may be proved that AB and BC are
> AC, 9
and BC and AC > AB. 10' Conc. & R. Therefore any two sides of a triangle, &c. Q.E.D.
by P. 9. Let AE bisect _ BAC. D. 1 by P. 16. ::: _ BEA is > _ EAC, and L CEA than EAB ; 2 P. 19.
BA is > BE, and AC than CE : 3 Conc. And BA + AC > BE + CE, which are equal to BC. 4 Rec. Therefore any two sides, &c.
Q.E.D. COR.—The difference of any two sides of a triangle is less than the rem. side. D. 11 by P. 20. The sides AC and BC are > AB ;
2 Sub. take AC both from (AC + BC,) and also from AB; 3 Ax. 5. .. BC is > AB diminished by AC, or than the difference
between AB and AC. APP.-1. Of all lines that can be drawn from one point, A, to another I, and reflected to a third point, B, those are shortest, AE, EB, which make the angle of incidence, AEG, equal to the angle of reflection, BED. C. 11 by P. 12 & Pst. 2. From B draw BD
perp. to DG,
B and produce
BDindefinitely; 2 P. 3.
make DC=DB, D
and join E, C. 3) Assum., Pst. 1. Also in DG as
AF, BF, and CF.
to DC, and the angles at D equal, 2 P. 4.
BE is = CE, and L. BED = L CED. 3) Sim.
In the same way we prove BF to be CF. 4 H. & C. & D. 2 Now _s BED, DEC, and AEG are all equal ; 5 C.
and AEC is a st. line: 6 D. 5 & C. 3, P. 20 | Also AFC is a A; ., AF and FC > AC: 71 D. 2, 3. but AC = AE and EB, and FB :
= FC; 8 Conc.
.:. AF, FC, or AF, FB, > AE and EB. 9) Rec.
Wherefore, of all lines that can be drawn &c. Q.E.D. 2. Natural causes act by the shortest lines; therefore all reflections are made by the lines which cause the angle of reflection to equal the angle of incidence. Hence, by means of a mirror placed horizontally, we may construct a triangle the perpendicular of which shall be representative of the height of any object.-See“ Euclid Practically Applied," Part I., Pr. 20, 1. p. 84.