Join CF, CG;
JFH, CF = HG, CG, ea. to ea.
HC common to As FHC, GHC;
LCHF = L CHG, and are adj. Ls,

and CH I AB.
Therefore, the I CH has been drawn, &c.

Prop. 7.

Def. 10.

PROP. XII. THEOR. 13. 1 Eu. The angles which one straight line makes with

another upon one side of it, are either two right angles, or are together equal to two right angles.

Let AB make with DC, on the same side of it, zs DBA, ABC; then / DBA + L ABC = 2 rt. Zs.

D

B

C

Def. 10.

Def. 10.

B C D
If _ ABC = _ DBA,

each of them is a rt. L; Prop. 10. But if _ ABC # DBA, draw BE 1 DC;

i. Ls CBE, DBE are rt. Zs.

LCBE = L ABC+ L ABE; add to these equals _ DBE;

L

DBE:
Again : _ DBA = < DBE + L ABE,

add to these equals ABC;

and ::

Ax. 2.

={_ ABBE: ABE + Ax. 2.

:: _DBA + LABC=

_DBE: but it has been shown that

CBE + LDBES same 3 Zs; :: Z CBE +_ DBE = _ DBA + LABC ; Ax. 1. but _ CBE + LDBE = 2 rt. 28,

.._DBA + LABC = 2 rt. Zs. Therefore the angles, &c.

Ax.l.

PROP. XIII. THEOR.

14. I Eu.

If at a point in a straight line, two other

straight lines, upon the opposite sides of it,
make the adjacent angles together equal to
two right angles, these two straight lines
shall be in one and the same straight line.

At the point B in AB let BC, BD, on the oppo, sides of AB, make the adj. <s ABC + ZABD = 2 rt. Zs; then shall BD be in the same straight line with BC.

If BD be not in the same str. line with CB, let BE be in the same str. line with it; then .: str. line AB makes with str. line CBE, on the same side of it, the Zs ABC, ABE: .. LABC + LABE = 2 rt. Zs;

Prop. 12.

Нур. .
Ax. 1.

Ax. 3.

but LABC + LABD = 2 rt. L;

:: LABC + LABE = L ABC+ LABD. From these equals take away _ ABC, .. [ ABE = L ABD;

greater, which is

absurd ; :. BE is not in the same str. line with BC. In like manner it may be shown, that no other line but BD can be in the same str. line with BC.

Wherefore, if at a point, &c.

or, less

PROP. XIV.

THEOR.

15. 1 Eu.

If two straight lines cut one another, the ver

tical or opposite angles shall be equal. Let the str. lines AB, CD, cut one another in E; then AEC = _ DEB, and _ CEB = L AED.

::str.line A Emakeswith CDthe Zs CEA,AED, Prop. 12.

LCEA + L AED = 2 rt. Zs.
Again,

str.line DEmakeswith ABthe Zs AED,DEB. Prop. 12. .. AED + L DEB = 2 rt. Zs,

:. CEA + L AED= LAED+ Z DEB.

Ax. 1.

Ax. 3.

From these equals take away the common

L AED, .. remain. _ CEA = remain. _ DEB. In the same manner it may be shown, that

Z CEB = ZAED. Therefore, if two straight lines, &c.

Cor. 1.-If two str. lines cut one another, the _s which they make at the pt. where they cut, are together equal to 4 rt. Zs.

Cor. 2.-All the angles made by any number of lines meeting in one pt., are together equal to 4 rt. _ s.

PROP. XV. THEOR.

16. I Eu.

If one side of a triangle be produced, the ex

terior angle is greater than either of the
interior opposite angles.

Let ABC be a A, and the side BC be prod. to D; then the ext. L ACD > L CBA or < BAC, the int. oppo. Z s.

{

AE =
BE =

Prop. 3.

Make EF = BE.

Join FC. Constr.

EC,

EF, Prop. 14.

2 АЕВ

= L CEF;

base AB = base CF, Prop. 4.

..

< BAC = L ECF, but / ACD > L ECF;

.. L ACD > L BAC. In the same way, if BC be bisected, and AC

be prod. to G, it may be shown, that Prop. 14.

_ BCG or < ACD > L ABC. Therefore, if one side, &c.

Ax. 9.

PROP. XVI. THEOR. 17. 1 Eu. Any two angles of a triangle are together less

than two right angles. Let ABC be any A, any two of its Ls are together less than 2 rt. Zs.

Produce BC to D. Prop. 15. : ext. _ ACD > L ABC, the int. and oppo. L ;

add ACB to each, .:: LACD + L ACB > LABC + L ACB; Prop. 12. but / ACD + L ACB = 2 rt. ZS,

:: L ABC + L ACB < 2 rt. Zs.