Const.

Def. 15.}

Prop. 7.

Def. 10.

Join CF, CG;

FH, CF = HG, CG, ea. to ea.
HC common to s FHC, GHC;

CHF CHG, and are adj. s, and CHAB.

Therefore, the CH has been drawn, &c.

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 s DBA, ABC; then DBA + 2

of it,
ABC

Def. 10.

If ABC

/ DBA,

each of them is a rt.

Prop. 10. But if▲ ABC ‡▲ DBA, draw BE DC;

Def. 10.

Ax. 2.

.. Zs CBE, DBE are rt.

and ::

s.

▲ CBE ABC+ 2 ABE; add to these equals / DBE;

:: ≤ CBE+/DBE=) < ABC + ≤ ABE+

Again

E={< LDBE:

DBA = DBE + ABE,

add to these equals ▲ ABC;

:: <DBA+2 ABC={< ABC + ABE + Ax. 2.

but it has been shown that

CBE+DBE

LDBE:

same 3 /s;

../CBE +2 DBE = /DBA+ ▲ ABC; Ax. 1. but CBE + ≤ DBE = 2 rt. s, :: LDBA +2≤ ABC = 2 rt.

Therefore the angles, &c.

s.

Ax. 1.

PROP. XIII. THEOR. 14. 1 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 + ABD 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 s ABC, ABE:

:: ZABC + ≤ ABE = 2 rt. Zs;

Prop. 12.

Hyp.

Ax. 1.

Ax. 3.

but

ABC + ▲ ABD = 2 rt. ;

:: ≤ ABC + / ABE = 2 ABC+≤ ABD. From these equals take away

ABC,

ABE

/ ABD;

or, less

greater, which is

.. 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.

PROP. XIV. THEOR. 15. 1 Eu.

If two straight lines cut one another, the vertical or opposite angles shall be equal.

Let the str. lines AB, CD, cut one another in E; then AEC = / DEB, and ▲ CEB AED.

=

...str.line AE makeswith CDthe /s CEA,AED, Prop. 12. CEA + ▲ AED 2 rt. S.

Again,

...str.lineDEmakes with ABthes AED,DEB.

Prop. 12... AED + 2 DEB = 2 rt. S,

Ax. 1. :: Z CEA + ▲ AED = LAED+ 2 DEB.

From these equals take away the common

.. remain.

▲ AED,

CEA = remain. ▲ DEB. In the same manner it may be shown, that CEBAED.

Therefore, if two straight lines, &c.

COR. 1.-If two str. lines cut one another, thes 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.

Ax. 3.

PROP. XV. THEOR.

16. 1 Eu.

If one side of a triangle be produced, the exterior 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. ▲ ACD > ▲ CBA or BAC, the int. oppo.

A

s.

F

< ACD > BAC.

Prop. 14.

ACD > < ECF;

In the same way, if BC be bisected, and AC be prod. to G, it may be shown, that

Z BCG or ACD >
Therefore, if one side, &c.

ABC.

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 s are together less than 2 rt.

A

s.