Nit LIS” SIRE Five atin y Meteors tit or 
eh ei op: XXVIi Turor. . Neaiiore 
ASS Pea asa SR tay ewe eile 3) gest 
The opposite sides of a parallelogram are equal, as 
well as the opposite angles. ' 
Draw the di BD, The tri s ADB, DBC 
have the pores DB; ig ea of the paral- 
lels AB, CD, theangle ABD = CDB, (2 Cor. 21.) and 
~ because of the parallels AD, BC, the angle ADB=DBC; 
therefore the triangles are equal, (1); and the sides 
AB, DC, which are opposite the equal angles, are equal. 
In like manner AD and BS are equal i, therefore the 
opposite sides of a parallelogram are equal. 
Again, from the equality of the triangles, it follows 
that the angle at A is equal to the ney a C; and it 
has been Rave that the angles ADB, BDC are re- 
ively equal to the angles CBD, DBA ; therefore 
whole angle ADC is equal to the whole angle 
ABC ; thus the opposite angles are equal, 
Cor, Two parallels AB, CD comprehended between 
two other parallels AD, BC are equal, 
Prop. XXVII. 
If in a quadrilateral ABCD, the opposite sides are 
equal, so AB=CD, and AD=BC ; the sides are 
parallel, and the figure is a parallelogram. 
For, drawing the diagonal DB, the triangles ABD, 
BDC have the three sides equal, each to each ; there- 
fore the angle ADB, opposite to the side AB, is equal 
to the angle CBD, opposite to the side CD, (11.) ; 
hence the side AD is parallel to BC, (1. Cor. 20.) For a 
like reason AB is parallel to CD; therefore the qua- 
drilateral ABCD is a parallelogram. 
Tueor. 
Par: XXVIII. Tueor. 
_E two opposite sides AB, CD of a quadrilateral are 
equal and el; the other two sides are equal and 
parallel, and the figure ABCD is a parallelogram. 
' For, having drawn the diagonal BD, since AB is pa- 
rallel to CD, the le ABD=BDC (2. Cor. 21.) ; be- 
sides the side AB=DC, and the side DB is common ; 
therefore the triangle ABDis equal to the triangle DBC, 
6.) ; therefore the side AD=BC, the angle ADB= 
BC; and consequently AD is parallel to BC; there- 
fore the figure ABCD is a parallelogram. 
SECT. IT. 
Or 4 Cincur. 
Definitions. 
1. The cireumference of a circle is a curve line, 
point of which is equally distant from a certain 
within it called the centre. 
bounded by that curve line. 
Note. Sometimes the circumference of a circle is call- 
ed the circle ; but it is xr to avoid ambiguity, by re- 
collecting that the circumference is.a line, and the circle 
a 
every 
point 
The circle is the space 
space. 
‘2, Every straight line, CA, CE, €D, &c. drawn from 
the centre to the circumference, is called a radius or 
semidiameter ; and every straight line, as AB, which 
GEOMETRY. 
205 
ses through the centre, and is terminated both ways 
By the circumference, is called a diameter. 
Cor. All the radii equal; also the diameters are 
all equal, and each is double of the radius, . 
5. An are of a circle is any portion of the circumfe- 
rence, as FHG, 
The chord or sublense of an arc is the straight line 
FG which joins its extremities, 
4. A segment is the space comprehended between an 
are and its chord. 
Note. The same chord FG corresponds to. two ares. 
FHG, FKG, also to two segments; but it is always, 
the least of the two that is meant, unless otherwise ex- 
r x 
, 5. A sector is a part of a circle comprehended by an. 
arc DE, and the two radii CD, CE drawn to its extre- 
mities, 
6.A straight line is said to be "oes in a circle, when 
its extremities are in the circumference, as FG. 
7. An angle is said to be ina segment of'a circle, 
when its vertex is on the arc of the segment, and the 
lines which contain it terminate in the extremities of 
the chord. 
8. A rectilineal figure is said to be inscribed in a circle, 
when all its angles are on the circumference of the 
circle. The circle is then said to be described about the 
figure. : 
9. A straight line is said to touch a circle, and is call- 
ed a tangent, when it, meets the circumference, and 
being produced does not cut it ; asthe line IKL. The 
point K in which the straight line meets. the circle is 
called the point of contact. 
10. Two circumferences of circles are said to touch 
each other, when they meet in one point only. 
11. A rectilineal figure is said to circumscribe a 
circle, when all its sides are tangents to the circumfe- 
rence ; the circle is then said to be inscribed in the 
figure. 
Prop. I... Tueor: 
Any diameter AB of a circle, divides the circumfe~ 
rence into two equal parts. 
For if the figure AEB be applied upon AFB, so that 
they may coincide in their common base AB ; it is ma- 
nifest that they must entirely coincide; for were it 
otherwise, some parts of the circumference would be 
farther from the centre than others, contrary to the de- 
finition of a circle. (Def. 1. Sect. 2.) 
Prop. II], TueEor. 
Every chord is shorter than the diameter. 
Of the 
Circle, 
Fig. 51. 
For if the radii AC, CD be drawn to the extremities Fig. 51. 
of the chord AD, then AD will be less than AC-+-CD 
(8. 1.) that is less than the diameter. 
Prop. III. Tueror. 
A straight line cannot cut a circle in more than two 
points. 
For if it could cut it in three, these would be. all 
equally distant from the centre; and so three equal 
straight lines might be drawn from the same point to 
terminate in the same straight line, which is unpossi- 
ble, (16. 1.) 
