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then the angle ADC b-ACD; as alfo, becaufe b 5. 1. BDc BC, the angle FDC-7 ACD. there- c fuppof fore is the angle FUCd ACD, that is, the d 9. ax. angle FDCADC. Which is impoffible.

3. Cafe. If D falls without the triangle ACB, let CD be joined.

ADC,

and the e 5.1. angle f 9. ax. BDC.

Again the angle ACD e
angle BCDe BDC. f Therefore the
ACD BDC, viz. the angle ADC
Which is impoffible. Therefore, &c.

PROP. VIII.

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If two triangles ABC, DEF, have two fides AB, AC equal to two fides DE, DF, each to each, and the base BC equal to the base EF, then the angles contained under the equal right lines shall be equal, viz. A to D.

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Because BC a EF, if the base BC be laid on a hyp. the bafe EF, bthey will agree: therefore whereas b ax. 8. AB c DE, and AC-DF, the point A will fall c hyp. on D (for it cannot fall on any other point, by the precedent propofition) and fo the fides of the angles A and D are coincident; d wherefore d 8. a thofe angles are equal. Which was to be Dem.

Coroll.

1. Hence, Triangles mutually equilateral, are alfo mutually e equiangular.

2. Triangles mutually equilateral, e are equal one to the other.

e 4. I.

PROP.

A

PROP. IX.

a 3. I.

D

b 1. 1.

B

F

c conftr.

d 8. I.

For AD c AE, and the base DF c

a 1. 1.

b 9.1.

c conftr.

d 4. I.

To bife, or divide into two equal parts, a right-lined angle given BAC. a Take AD to AE, and draw the line DE; upon which b make an equilateral triangle DFE. draw the right line AF; it shall bifect the angle.

and the fide AF is common, FE. d therefore the angle DAFEAF. Which was to be done.

Coroll.

Hence it appears, how an angle may be cut into any equal parts, as 4, 8, 16, &c. to wit, by bifecting each part again.

The method of cutting angles into any equal parts required, by a Rule and Compass, is as yet unknown to Geometricians.

D

PROP. X.

To bifet a right line given AB.

Upon the line given AB a erect an equilateral triangle ABC; and b bifect the angle C with the right line CD. That line fhall also bisect the Bline given AB.

For ACc BC, and the fide CD is common, and the angle ACDC-BCD.therefore ADd BD, Which was to be done.

The practice of this and the precedent Propofition is eafily fhewn by the conftruction of the I Prop, of this Book.

PROP

PROP. XI.

From a point C in a right line given AB to erect a right line CF at right angles.

a Take on either fide

of the point given CDa 3. 1. =CE. upon the right

AD C EB line DE b erect an equi-b 1. 1. lateral triangle, draw the line FC, and it will be the perpendicular required.

For the triangles DFC, EFC are mutually ce-c conftr. quilateral; d therefore the angle DCF-ECF. ed 8. I. therefore FC is perpendicular. Which was to be done.e 10. def. The practice of this and the following is eafily performed by the help of a fquare.

AE

PROP XII.

Upon an infinite right line given AB, from a point given that is not in it, to let fall a perpendicular

FB right line CG.

From the center

Ca defcribe a circle cutting the right line given

AB in the points E and F. Then bifect EF in a 3. post. G, and draw the right line CG, which will be b 1o. 1. the perpendicular required.

Let the lines CE, CF be drawn. The triangles

EGC, FGC are mutually c equilateral. d there-c conftr. fore the angles EGC, FGC are equal, and by d 8. I. e confequence right. e Wherefore GC is a per-e 10. def. pendicular. Which was to be done.

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a def. 10.

br. 1.

c 19. ax.

d 3. ax. e 2. ax.

a 13. 1. b hyp. €9.ax.

If the angles ABC,ABD be equal, a then they make two right angles; if unequal, then from the point Bb let there be erected a perpendicular BE. Because the angle ABC cto a right+ ABE, and the angles ABD d to a right - ABE, therefore fhall be ABC ABD e to two right angles ABE - ABE two right angles. Which was to be demonftrated.

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Corollaries.

1. Hence, if one angle ABD be right, the other ABC is alfo right; if one acute, the other is obtule, and fo on the contrary.

2. If more right lines than one ftand upon the fame right line at the fame point, the angles fhall be equal to two right.

3. Two right lines cutting each other make angles equal to four right.

4. All the angles made about one point, mak four right; as appears by Coroll.. PROR. XIV.

B

If to any right line AB, and a point therein B, two right lines, not drawn from the fame Jide, do make the angles ABC, 4BD on each fide Dequal to two right, the lines CB,BD fhall make one ftrait line.

If you deny it, let CB, BE make one right line, then fhall be the angle ABC+ABE a two right angles bABC+ABD. Which is abfurd.

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If to any right line GH, and in it a point A, two right lines being drawn EA, AF, and not taken on the fame fide,make the vertical (or oppofite) angles Dand B equal, thofe right lines EA, AF, do meet directly and make one ftrait line. For two right angles are a equal to the angle a 13. 1. b 14. I. D+A a B A. b therefore, EA, AF, are in a ftrait line. Which was to be demonftrated.

Schol.

If four right lines EA, EB, EC, ED, proceeding from one point E, make the angles vertically oppofite equal the one to the D other, each two lines, AE, Ꭼ Ᏼ, and CE, ED, are placed in one ftrait line. For because the angle AEC AED - CEB DEB a to 4 right angles,therefore the angie AECAEDb- CEB DEB = to two a 4. c.13.1 right angles, c therefore CED and AEB are2x. ftrait lines. Which was to be demonftrated.

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PROP. XVI.

One fide BC of any triangle ABC being produc'd, the outward angle ACD will be greater than either of the inward and oppofite angles CA B, LCBA.

Let the right lines AH, BE

b hyp.and

C 14. I.

a bifect the hides AC, BC;a 10.1. &
produce EF BE, and H1,1. poft.
b=b 3. I

B

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