micircle, fince CD, a Line paffing through the Center B, is a Diameter, therefore each of the parts AC, AD is a Quadrant, i.e. 90 Degrees; fo the measure of a right Angle is always 90 Degrees. 32. If one Line AB fall any way upon another,

C B D

CD, then the Sum of the two Angles ABC, ABD is always equal to the Sum of two right Angles. For on the point B, defcribing the Circle CAD, it is plain, that CAD is a Semicircle (by '15th); but CAD is equal to CA and AD the measures of the two Angles; therefore the Sum of the two Angles is equal to a Semicircle, that is, to two right Angles (by the last).

Cor. 1. From whence it is plain, that all the Angles which can be made from a point in any Line, towards one fide of the Line, are equal to two right Angles.

2. And that all the Angles which can be made about a Point, are equal to four right ones. 33. If one Line AC crofs another B D in the Point E, then the oppofite Angles are equal, viz. BEA to CED, and BEC equal to AED. For upon the point E, as a Center, describing the Circle ABCD, it is plain ABC is a Semicircle, as alfo

B

E

A

D

BCD (by 15th) therefore the Arch ABC is equal to the Arch BCD; and from both taking the com mon Arch BC, there will remain AB equal to CD, i. e. the Angle BEA equal to the Angle CED (by Art. 30.). After the fame manner we may prove, that the Angle BEC is equal to the Angle AED.

34. Lines which are equally diftant from

one another are called Parallel Lines; as AB, CD.

A

C

B

D

35. If a Line GH cross two Parallels A B, CD, then the external Angles are equal, viz. GEB equal to CF H and AEG equal to HFD. For fince A B and CD are parallel to one another, they may be confidered as one broad Line, and GH croffing it; then the vertical or oppofite Angles GEB, CFH are equal (by the 33d) as alfo AEG and HFD by the fame.

36. If a Line G H cross two Parallels A B, CD then the alternate An

G

gles, viz. A EF and

EFD, or CFE and

E

FEB are equal; that is, the Angle A EF

A

-B

is equal to the Angle C

D

EFD, and the An

gle CFE is equal to

the Angle FEB, for

F

H

GEB is equal to AEF (by the 33d.) and CF His equal to EFD by the fame, but GEB is equal to CFH by the last. Therefore AEF is equal to EFD; the fame way we may prove FEB equal to EFC.

37. If a Line G H cross two parallel Lines A B, CD, then the external Angle GEB is equal to the internal opposite one EFD, or GEA equal to CFE. For the Angle AEF is equal to the Angle EFD by the laft; but A EF is equal GEB (by the 33d) therefore GEB is equal to E FD; the faine way we may prove AEG equal to CFE.

38. If a Line GH cross two parallel Lines AB, CD, then the Sum of the two internal Angles, viz. BEF and DFE, or AEF and CFE are equal to two right Angles; for fince the Angle GEB is equal to the Angle EFD (by the laft) to both add the Angle FEB, then GEB and BEF are equal to BEF and DFE; but GEB and BEF are equal to two right Angles (by the 32d) therefore BEF and DFE are alfo equal to two right Angles. The fame way we may prove that AEF and CFE are equal in two right Angles.

39. A Figure is any part of Space bounded by Lines or a Line. If the bounding Lines be ftreight, it is called a Rectilineal Figure as A; if they be curved, it is called a curvilineal Figure as B or C ; if they be partly curve Lines and partly streight, it is called a mixt Figure as D.

A

40. The most fimple rectilineal Figure is that which is bounded by three right Lines, and is called a Triangle, as A.

41. Triangles are divided into different kinds, both with refpect to their Sides and Angles with refpect to their Sides they are commonly divided into three kinds, viz.

42. A Triangle having all it's three Sides equal to one another, is called an Equilateral Triangle, as A. 43. A Triangle having two of it's Sides equal to one another, and the third Side not equal to either of them, is called an Ifofceles Triangle, as B.

44. A Triangle having none of it's Sides equal to one another, is called a Scalene Triangle, as C.

A A A

45. Triangles, with refpect to their Angles, are divided into three different kinds, viz.

46. A Triangle having one of it's Angles right, is called a Right-Angled-Triangle, as A.

47. A Triangle having one of it's Angles obtuse, or greater than a right Angle, is called an ObtuseAngled-Triangle, as B.

48. Laftly, a Triangle having all it's Angles acute, is called an Acute-Angled-Triangle, as C.

A

49. In all right angled Triangles, the Sides comprehending the right Angle are called the Legs, and the Side oppofite to the right Angle is called the Hypothenufe. Thus in the right angled Triangle ABC (the right B

C

Angle being at B) the two Sides A B and BC which comprehend the right Angle ABC, are the Legs of the Triangle, and the Side AC, which is oppofite to the right Angle ABC, is the Hypothenuse of the right-angled-Triangle ABC.

50. Both obtufe and acute angled Triangles are in general called Oblique-Angled-Triangles; in all which any Side is called the Bafe, and the other two the Sides.

1

the Vertex, viz. The Angle oppofite to the Bafe; and if from A you draw the Line A D perpendicular to BC, then the Line AD is the Height of the Triangle ABC fanding on BC as it's Bafe.

Hence all Triangles ftanding between the fame Parallels have the fame Height, fince all the Perpendiculars are equal by the Nature of Parallels.

Аг

A

B

C

52. A Figure bounded by four Sides is called a Quadrilateral or Quadrangular Figure, Das ABDC.

53. Quadrilateral Figures whofe oppofite Sides are parallel, are called Parallelograms. Thus in the quadrilateral Figure ABDC, if the Side AC be parallel to the Side BD which is oppofite to it, and AB be parallel to CD, then the Figure ABDC is called a Parallelogram.

54. A Parallelogram having all it's Sides equal and Angles right, is called a Square; as A.

55. That which hath only the oppofite Sides equal and it's Angles right, is called a Rectangle; as B. 56. That which hath equal Sides but oblique Angles, is called a Rombus, as C; and is juft an inclin'd Square.